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Engineering Structures 29 (2007) 2641–2653www.elsevier.com/locate/engstruct

Wind load effects and equivalent static wind loads of tall buildings based onsynchronous pressure measurements

Guoqing Huang, Xinzhong Chen∗

Wind Science and Engineering Research Center, Department of Civil and Environmental Engineering, Texas Tech University, Lubbock, TX 79409, USA

Received 20 September 2006; received in revised form 9 January 2007; accepted 10 January 2007Available online 1 March 2007

Abstract

This paper addresses the wind load effects and equivalent static wind loads (ESWLs) of tall buildings based on measured synchronous surfacepressures in a wind tunnel. The variations of the gust response factors (GRFs) associated with different alongwind responses are studied, pointingout the deficiency of the traditional ESWL modeling based on the GRF associated with the top displacement. The higher mode contributionsto various building responses such as top acceleration, story shear force and bending moment at different building elevations are investigated,which reveals the noticeable contributions of higher modes to some important building responses. The comparison of wind load effects withthose based on the wind load specification ASCE 7-05 is also conducted. Furthermore, the practical methodology of modeling ESWLs on tallbuildings involving the influence of higher mode contributions is proposed. Finally, using the spatiotemporally varying wind load information, themode shape correction factors required in the high frequency force balance (HFFB) technique are revisited to examine the efficacy of empiricalformulations adopted in current practice.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Wind; Wind load; Tall building; Structural dynamics; Wind tunnel test; Aerodynamics

1. Introduction

Wind loads on tall buildings can be quantified throughmultiple point synchronous scanning of pressures on a buildingmodel surface in a wind tunnel, or by high frequency forcebalance (HFFB) measurements, or by simplified wind loadingcodes. The synchronous pressure measurements provide adetailed description of spatiotemporally varying wind loads,while the HFFB measurements offer an estimate of thegeneralized forces of the fundamental modes of vibration. Thewind loading codes give simplified and often conservativeequivalent static wind loads (ESWLs) on isolated buildingswith simple geometric configurations. For tall buildingswith uncoupled mode shapes in primary directions, wind-induced responses in each primary direction can be analyzedindependently often involving only the fundamental mode.

The fundamental mode contribution dominates globalbuilding response such as the top displacement and base

∗ Corresponding author. Tel.: +1 806 742 3476x324; fax: +1 806 742 3446.E-mail address: [email protected] (X. Chen).

0141-0296/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2007.01.011

bending moment. However, higher modes may have noticeablecontributions to some responses such as the top acceleration.The studies by Simiu [19], Kareem [15] and Simiu andScanlan [20] using analytical loading models demonstrated thatthe contributions from higher modes may reach to about 20%of the top acceleration. More detailed discussions concerningthe contributions of higher modes to various building responsesusing measured spatiotemporally varying wind loads have notbeen adequately addressed in the literature.

Current design practice often requires the dynamic windloads to be represented in terms of the ESWLs. The gustresponse factor (GRF) approach proposed by Davenport [8]has been widely used in design codes and standards worldwidefor modeling alongwind loading. This approach leads to anESWL given by the mean wind load multiplied by a GRF, oftenassociated with the top displacement. Studies have shown thatGRF may vary widely for different response (e.g., Holmes [12];Chen and Kareem [4–6]). Therefore, using a single GRFfor all response components may lead to remarkable over-or underestimates of building response. Moreover, the GRFapproach falls short in providing physically meaningful ESWLs

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2642 G. Huang, X. Chen / Engineering Structures 29 (2007) 2641–2653

for the acrosswind and torsional responses, which are typicallycharacterized by zero mean wind loading and response. Itshould be mentioned that the Australian/New Zealand windloading standard AS/NZS 1170.2: [2] has adopted an advancedESWL modeling with a dynamic response factor whichincreases with increasing height [2]. The advanced modelingof ESWLs on tall buildings including background and resonantcomponents has been addressed extensively in the literature(e.g., Kasperski [17]; Holmes [13]; Kareem and Zhou [16];Chen and Kareem [4–6]; Repetto and Solari [18]).

The HFFB technique requires empirical mode shapecorrection factors for estimating the fundamental generalizedforce through measured base bending moment or base torque.A number of empirical formulas have been suggested in theliterature (e.g., Vickery et al. [23]; Boggs and Peterka [3]; Xuand Kwok [25]; Ho et al. [10]; Holmes [11]; Holmes et al. [14];Chen and Kareem [4]). HFFB measurements have also beenused to identify spatiotemporally varying fluctuating wind loadsbased on assumed loading models (e.g., Yip and Flay [26]; Xieand Irwin [24]). The detailed spatiotemporally varying pressuredata provide an opportunity to verify the adequacy of theseempirical formulas used in practice.

This study addresses the wind load effects and ESWLs of tallbuildings based on measured surface pressures of 20- and 50-story building models. The variations of the GRFs and highermode contributions for various building responses are studied indetail. Furthermore, the comparison of building response withthat based on wind loads specified in ASCE 7-05 (ASCE [1])is conducted. Finally, based on the detailed dynamic wind load,the mode shape correction factors used in the HFFB techniqueare revisited to examine the efficacy of empirical formulasadopted in current practice.

2. Analysis framework

Consider the wind-excited response of a tall building inone of three primary directions in which the building exhibitsa one-dimensional uncoupled mode shape of vibration. Thewind load acting at each floor level, denoted by the mean anddynamic components P̄j and Pj (t) ( j = 1, 2, . . . , N ; N is thetotal number of building floors), are quantified from multiplepoint synchronous scanning of pressures on the building modelsurface in a wind tunnel. The mean component of a specificresponse of interest, R, e.g., displacement, shear force, bendingmoment and member force, is estimated by static analysis underthe mean wind load P̄j and expressed as

R̄ =

N∑j=1

µ j P̄j (1)

where µ j is the influence function of response R defined as theresponse under a unit load acting at the j th floor.

The dynamic response, R(t), is determined through modalanalysis:

R(t) =

N∑n=1

Bnqn(t) (2)

Mn[q̈n(t) + 2ξnωn q̇n(t) + ω2nqn(t)] = Qn(t) (3)

where Bn is the modal participation coefficient; qn(t) is thegeneralized displacement; Mn, ξn , and ωn are the generalizedmass, damping ratio and circular frequency, respectively;Qn(t) =

∑Nj=1 Θ jn Pj (t) is the generalized force; and Θ jn is

the mode shape in terms of the j th floor motion.The modal contributions to the response can be divided

into two parts: the first Nd modes where the dynamiceffect is significant, and modes Nd + 1 to N with naturalfrequencies such that their response is essentially quasi-static.Subsequently, the classical mode displacement superpositionmethod given by Eq. (2) can be replaced by the static correctionmethod (e.g., Chopra [7]):

R(t) =

Nd∑n=1

Bnqn(t) +

N∑n=Nd+1

Bnqnb(t)

=

Nd∑n=1

Bnqnr (t) +

N∑n=1

Bnqnb(t) (4)

where qnb(t) = Qn(t)/Kn is the nth background (quasi-static)generalized displacement and qnr (t) = qn(t) − qnb(t) is thenth resonant generalized displacement, i.e., the generalizeddisplacement excluding the quasi-static component; Kn =

Mnω2n is the nth generalized stiffness.

It is noted that the static correction method essentiallycorresponds to the analysis approach of separating the dynamicresponse into the background and resonant components. Thisapproach has traditionally been introduced in the frequencydomain, and widely used in the analysis of wind-inducedresponse of structures. The quantification of the backgroundresponse including all mode contributions is equivalent to thequasi-static analysis in terms of the influence function, i.e.,

Rb(t) =

N∑j=1

Bnqnb(t) =

N∑j=1

µ j Pj (t). (5)

Response analysis can be carried out in either time orfrequency domain. In the time domain scheme, the responsetime history is quantified by using the step-by-step integrationmethod, e.g., Newmark’s numerical method. Based on theresponse time history, R(t), the root-mean-square value (RMS),σR , and the peak value, Rmax, can be directly determined.The ensemble-averaged quantities are then computed based onmultiple samples. The peak factor and gust response factorassociated with R(t) are given as gR = Rmax/σR and G R =

(R̄ + Rmax)/R̄, respectively.Alternatively, when the frequency domain scheme is

utilized, the RMS value of response R(t) is quantified bycombining the background and resonant components as

σ 2R = σ 2

Rb+ σ 2

Rr(6)

σ 2Rb

=

N∑j=1

N∑k=1

µ jµkσPj σPk rPjk (7)

G. Huang, X. Chen / Engineering Structures 29 (2007) 2641–2653 2643

σ 2Rr

=

Nd∑m=1

Nd∑n=1

Bm Bnσqmr σqnr rmnr (8)

where σRb and σRr are the RMS values of the background andresonant response components, respectively; σPj is the RMSvalue of Pj (t); rPjk is the correlation coefficient between Pj (t)and Pk(t); σqnr is the RMS value of the resonant generalizeddisplacement in the nth mode and can be estimated using theclosed-form formulation:

σqnr =1

Kn

√π fn

4ξnSQn ( fn) (9)

where SQn ( f ) is the power spectral density (PSD) functionof the generalized force Qn(t); and rmnr is the correlationcoefficient between the mth and nth resonant modal responses,which depends on not only the frequency ratio and dampingratios, but also the coherence of the generalized forces, and canbe estimated as follows [9,5,6]:

rmnr = ρmnrαmnr (10)

ρmnr

=8√

ξmξn(βmnξm + ξn)β3/2mn

(1 − β2mn)2 + 4ξmξnβmn(1 + β2

mn) + 4(ξ2m + ξ2

n )β2mn

(11)

αmnr = Re[SQmn ( f )]/√

SQm ( f )SQn ( f )

∣∣∣f = fm or fn

(12)

where βmn = fm/ fn ; and Re denotes the real part of thecomplex value.

The accuracy of estimating resonant response throughthe closed-form formulations instead of using the numericalintegration of response PSD will be affected by the smoothnessof the PSD of the generalized force around the modal frequency.The PSD of the generalized force calculated from wind loadinghistory with limited samples and time duration are often ajagged function of frequency. To enhance the accuracy of thepredicted resonant response, a “smooth” estimation of the PSDat the modal frequency can be used in the analysis by averagingthe spectra over the half-power bandwidth.

If the background response is quantified through modalanalysis, similar to the resonant response, the completequadratic combination (CQC) should be used:

σ 2Rb

=

N∑m=1

N∑n=1

Bm Bnσqmbσqnbrmnb (13)

σqnb =1

Kn

√∫∞

0SQn ( fn)d f =

σQn

Kn(14)

rmnb = rQmn (15)

where σqnb and σQn are the RMS vales of the nth backgroundgeneralized displacement and generalized force; rmnb and rQmn

are the correlation coefficients between generalized backgrounddisplacements and between generalized forces in the mth andnth modes.

The peak dynamic response including the background andresonant components is:

Rmax =

√g2

bσ 2Rb

+ g2r σ 2

Rr(16)

where gb and gr are the peak factors for the background andresonant components, respectively, usually ranging from 3 to 4.

The building acceleration is of interest for the buildinghabitability design, which can be determined through modalanalysis in either time or frequency domain. When thefrequency domain approach is employed, the RMS values ofthe background and resonant components of the nth generalizedacceleration are determined as

σq̈nb =1

Kn

√∫∞

0(2π f )4SQn ( f )d f (17)

σq̈nr = ω2nσqnr =

1Mn

√π fn

4ξnSQn ( fn). (18)

The background acceleration is generally negligibly small ascompared to the resonant component. When only the resonantcomponent is considered, the RMS value of the acceleration atthe j th floor level, i.e., a j (t) =

∑Nn=1 Θ jn q̈nr (t), is given by

σ 2a j

=

N∑m=1

N∑n=1

Θ jmΘ jnσq̈mr σq̈nr rmnr . (19)

3. Building examples and associated wind loading

3.1. Building dynamic properties

Two shear buildings of 20 and 50 stories with the samesquare cross section of 40 m × 40 m and story height of4 m are chosen as examples. Both buildings are modeledas a lumped-mass system with the floor mass of 1.228 ×

106 kg that corresponds to a building density of 192 kg/m3.Both buildings have uncoupled mode shape in each primarydirection, thus building response in each direction can beanalyzed independently. The fundamental frequencies in threeprimary directions are assumed to be identical, whereas thetorsional frequency is generally larger than the translationalfrequency. The fundamental frequencies for the 20- and 50-story buildings are assumed as 0.4415 Hz and 0.2122 Hz,respectively, according to the empirical formula for the naturalfrequency of steel moment resisting frames suggested inASCE 7-05. Four different story stiffness distributions for eachbuilding are considered. In the first three cases, the storystiffness is assumed to vary over the building height whichis determined based on the prescribed fundamental frequencyand modal shape. The fundamental mode shape is assumedto follow a power law with an exponent β = 1.0, 1.25, and1.5, respectively. The last case corresponds to a uniformly-distributed story stiffness which is determined from the givenfundamental frequency. Fig. 1 shows the mode shapes of thefirst 5 modes of the 50-story building. The modal frequenciesof both buildings are found to be well separated at least in


Fig. 1. Mode shapes of the 50-story building.

(a) 20-story building. (b) 50-story building.

Fig. 2. Experimental building models with pressure taps.

the first 10 modes. For the example of the 50-story buildingwith β = 1.25, the frequency ratios are f2/ f1 = 2.3 andf3/ f2 = 1.6. The damping ratio for all modes of both buildingsis assumed to be 2%.

3.2. Wind loading

Wind loads on both buildings are derived from multiplepoint synchronous scanning of pressures on the building modelsurface in a wind tunnel [22]. These experiments were carriedout with a length scale 1/400 for a suburban terrain where thepower law exponent of the mean wind speed profile was 1/6.Total 200 and 500 wind pressure taps shown in Fig. 2 wereuniformly distributed over the four wall surfaces of the modelsof 10 cm × 10 cm × 20 cm and 10 cm × 10 cm × 50 cm for the20- and 50-story buildings, respectively. Each wall of these twomodels has 10 and 25 layers of pressure taps, respectively. Thewind direction was set normal to the wall face.

The sampling interval of wind pressure time history was0.00128 s, and total 32 768 data were recorded during the time


Fig. 3. Story force coefficients.

duration of about 42 s. As the mean wind speed at the buildingtop, UH , varies from 20 to 60 m/s, the full-scale duration ofthe record ranges from 148 to 50 min for the 20-story building.To maintain the same mean wind speed profile for the 50-storybuilding, the range of UH is set from 23.3 to 69.9 m/s andthe corresponding time duration is from 160 to 53 min. Theentire pressure data were divided equally into six independentsamples to compute the structural responses which are thenused for estimating their ensemble-averaged quantities.

The translational forces and torque at each floor level aredetermined by integrating wind pressures within the tributaryarea. The forces of the stories at which there is no pressure taplocated are determined by interpolating the story forces actingon two adjacent stories. Fig. 3 shows the mean and RMS storyforce coefficients in the three primary directions of the 20- and50-story buildings. The translational force coefficient is definedas the ratio of the force to the product of wind speed pressureat the building top and the building frontal area of each story,


(a) Fundamental mode. (b) Second mode.

Fig. 4. PSDs of the generalized forces (50-story building, β = 1.25).

Fig. 5. RMS values of the generalized forces (50-story building, β = 1.25).

whereas the torque coefficient is calculated by further dividingthe building width. It is observed that the mean alongwindforce coefficient increases along the building elevation, whilethe mean acrosswind story force and torque coefficients arenegligibly small. The RMS force coefficients in three primarydirections are almost constant over the building height. Figs. 4and 5 show the PSDs and RMS values of the generalized forcesassociated with the fundamental and second modes in threedirections. The mode shapes are normalized according to unitygeneralized masses. It is noted that the acrosswind dynamicload is greater than the alongwind and torsional loads. Thefundamental modes correspond to larger loads than those of thehigher modes.

4. Results and discussions

4.1. Mean, dynamic responses and GRF

As the buildings with four different story stiffnessdistributions show similar characteristics of wind load effects,only the building with the power law exponent of thefundamental mode shape β = 1.25 is discussed here. Fig. 6shows the alongwind displacement, shear force and bendingmoment at different building elevations of the 50-story buildingat UH = 46.6 m/s calculated from the time history analysis.The RMS and peak values of the acrosswind and torsionalresponses follow similar variations along the building elevationas the alongwind response. The peak factor of the response

ranges between 3.5 and 4.0. The analysis in the frequencydomain is also carried out which leads to agreeable RMSresponse with an error of less than 10%.

Fig. 7 shows the RMS values of the alongwind background,resonant and total dynamic top displacement of both buildingsat varying wind speeds. The building response increases withwind speed approximately following a power law. The powerlaw exponent of the background response is 2.0. The resonantresponses of the 50-story building in alongwind, acrosswindand torsional directions have an exponent of about 2.9, 3.5 and2.2, respectively. Consequently, the total dynamic responses inthree directions correspond to an exponent of about 2.4, 3.0, and2.1, respectively. The top accelerations in three directions havea power law exponent of 2.5–3.4. The acrosswind accelerationis significantly greater than those in the other two directions.The power law exponent of resonant response is greater than2.0, because in addition to the wind load proportional to thewind speed square, the decrease in the modal reduced frequencyas wind speed increases introduces extra wind load. As theresonant response increases with wind speed at a faster ratethan the background response, the ratio of the backgroundto resonant response decreases with increasing wind speed. Itshould be noted that the level of modal damping ratio stronglyinfluences the resonant response and thus the ratio of thebackground to resonant component.

The variations of the GRFs associated with differentalongwind responses of both buildings obtained from the timedomain analysis are shown in Fig. 8. The GRFs for the topdisplacement, base shear force and base bending moment arealmost same and about 2.0. The GRFs for the shear forces andbending moments at higher floor levels are remarkably larger.Moreover, GRFs for the shear force and bending moment atthe same level are almost identical. As pointed in Holmes [12]and Chen and Kareem [4], the alongwind building response athigher floor levels will be significantly underestimated usingthe ESWL given as the mean wind load multiplied by theGRF associated with the top displacement or base bendingmoment as suggested in most current wind loading codes andspecifications. The variations of GRFs over building heightfollow similar features at different wind speeds. As the dynamicresponse grows with wind speed at a faster rate than the static


Fig. 6. Alongwind displacement, shear force and bending moment (50-story building, β = 1.25, UH = 46.6 m/s).


Fig. 7. Alongwind top displacement at varying wind speeds (β = 1.25).

(a) 20-story building(UH = 40 m/s).

(b) 50-story building(UH = 46.6 m/s).

Fig. 8. GRFs for different alongwind responses (β = 1.25).

response, the GRF slightly increases with increasing windspeed as shown in Fig. 9.

4.2. Higher modal contributions

Tables 1 and 2 show the influence of higher modetruncation on the base and top responses of the 50-storybuilding at UH = 46.6 m/s. Figs. 10 and 11 show theinfluence of higher mode truncation on the 50-story building

response expressed as the ratios√

σ 2R1b

+ σ 2R1r

/

√σ 2

Rb+ σ 2

Rr

and√

σ 2Rb

+ σ 2R1r

/

√σ 2

Rb+ σ 2

Rr. These ratios for the 20-story

building are shown in Figs. 12 and 13. The difference betweenFigs. 10 and 11, and between Figs. 12 and 13, is in theestimation of the background response. Figs. 10 and 12 showthe influence of truncating higher mode contributions to bothbackground and resonant components, while Figs. 11 and 13are those by truncating higher mode contributions only to theresonant component.

As the background modal response may have noticeablenegative correlation, considering only the fundamental modemay not necessarily underestimate background response. Whenthe frequency domain modal analysis procedure is followed,the CQC scheme should be used for estimating the backgroundresponse especially for lower buildings where the backgroundcomponent is more dominant. In such cases, the analysis of



Fig. 9. GRFs at varying wind speeds (β = 1.25).

Table 1Influence of higher mode truncation on translational response (50-story building, β = 1.25, UH = 46.6 m/s)

Top displacement Shear force Bending moment Top accelerationTop Base Top Base

Alongwind

σR1b /σRb 1.09 2.47 0.76 2.47 0.96 –σR1r /σRr 1.00 0.81 0.99 0.81 1.00 0.81σRb /σRr 0.73 0.26 1.04 0.26 0.83 –√

σ 2R1b

+ σ 2R1r

/√

σ 2Rb

+ σ 2Rr

1.03 1.00 0.87 1.00 0.99 0.81√σ 2

Rb+ σ 2

R1r/√

σ 2Rb

+ σ 2Rr

1.00 0.82 0.99 0.82 1.00 0.81

Acrosswind

σR1b /σRb 1.16 3.94 0.72 3.94 0.96 –σR1r /σRr 1.00 0.93 1.00 0.93 1.00 0.92σRb /σRr 0.44 0.12 0.70 0.12 0.50 –√

σ 2R1b

+ σ 2R1r

/√

σ 2Rb

+ σ 2Rr

1.03 1.03 0.92 1.03 0.99 0.92√σ 2

Rb+ σ 2

R1r/√

σ 2Rb

+ σ 2Rr

1.00 0.93 1.00 0.93 1.00 0.92

Table 2Influence of higher mode truncation on torsional response (50-story building, β = 1.25, UH = 46.6 m/s)

Top rotation Torque Top angular accelerationTop Base

σR1b /σRb 1.15 1.99 0.72 –σR1r /σRr 0.99 0.64 0.96 0.64σRb /σRr 0.39 0.15 0.61 –√

σ 2R1b

+ σ 2R1r

/√

σ 2Rb

+ σ 2Rr

1.01 0.69 0.91 0.64√σ 2

Rb+ σ 2

R1r/√

σ 2Rb

+ σ 2Rr

0.99 0.65 0.97 0.64

the background response directly using the influence functionis computationally more efficient and thus recommended. Forvery tall buildings, the background response becomes lessimportant and thus can be approximately estimated by themodal analysis including only the fundamental mode.

In the case of resonant response, as the modal frequenciesof both buildings are well separated, the correlation betweenresonant modal responses is negligible. Accordingly, neglecting

the higher mode contributions always results in lower resonantresponse. Table 3 summarizes the contributions of first fivemodes to the alongwind resonant responses of the 50-storybuilding, expressed in terms of the ratio of the RMS modalresponse to the total RMS response. It is observed that thehigher mode contributions have different influence on differentresponses. That is attributed to their different values of modalparticipation coefficients. Compared to the top displacement,


(a) Alongwind. (b) Alongwind. (c) Alongwind. (d) Torsional.

Fig. 10. Influence of higher mode truncation in both background and resonant responses (50-story building, β = 1.25, UH = 46.6 m/s).


Fig. 11. Influence of higher mode truncation in the resonant response (50-story building, β = 1.25, UH = 46.6 m/s).

Table 3Mode contributions to alongwind resonant response (50-story building, β = 1.25, UH = 46.6 m/s)

Mode number Top displacement Shear force Bending moment Top accelerationTop Base Top Base

1 1.00 0.81 0.99 0.81 1.00 0.812 0.09 0.43 0.15 0.43 0.03 0.433 0.02 0.28 0.06 0.28 0.01 0.284 0.01 0.18 0.03 0.18 0.00 0.185 0.00 0.13 0.02 0.13 0.00 0.13

and base shear force and bending moment, the story forcesat upper floor levels are more influenced by the higher modecontributions. For the top acceleration of the 50-story building,considering only the fundamental mode leads to an error ofabout 19% in alongwind direction, 8% in acrosswind directionand 36% in torsion. These results are consistent with those

reported in the literature [19,15,20]. It should be emphasizedthat the significance of higher mode contributions dependson both structural dynamics and wind loading characteristics,particularly on the PSDs of the generalized forces at themodal frequencies. In the case of the 20-story building,considering only the fundamental mode leads to the predicted



Fig. 12. Influence of higher mode truncation in both background and resonant responses (20-story building, β = 1.25, UH = 40 m/s).


Fig. 13. Influence of higher mode truncation in the resonant response (20-story building, β = 1.25, UH = 40 m/s).

top acceleration with an error of about 30%, 37% and 39% inalongwind, acrosswind and torsional directions, respectively.

Figs. 11 and 13 also demonstrate that story forces at upperfloor levels will be significantly underestimated when onlythe fundamental mode is involved in the resonant responseand the background response is estimated by the quasi-staticanalysis. The influence of higher mode truncation on resonantcomponent is almost insensitive to the variation of wind speedas shown in Fig. 14. When both background and resonantresponse components are estimated by only considering thefundamental mode as shown in Figs. 10 and 12, the dynamicresponse may be over- or underestimated, particularly forlower buildings where the background component has moresignificant influence on the total dynamic response. It isnoted that the top displacement and base bending moment areless sensitive to higher mode contributions. In particular, forbuildings with a linear fundamental mode shape, the higher

modes will provide no contribution to the base bending momentdue to the orthogonality between the higher mode shape and theinfluence function, although they would affect other responsecomponents.

4.3. Modeling of ESWLs

When only the fundamental mode contribution is consid-ered, the ESWLs in three primary directions can be defined bydistributing the base bending moment or base torque follow-ing the fundamental inertial force distribution (e.g. Chen andKareem [4]). This concept can also be used in the cases withconsideration of higher mode contributions by further intro-ducing modification factors as adopted in current seismic load-ing specifications (e.g., Chopra [7]). However, considering thedistinct influence of higher modes on the background and reso-nant responses, it is more convenient and physically meaningful


(a) 20-story buildings. (b) 50-story buildings.

Fig. 14. Influence of higher mode truncation in the resonant response at varying wind speeds (β = 1.25).

to model the background and resonant ESWLs (BESWL andRESWL) separately, especially when the background responseis noticeable.

The BESWL can be modeled using the load responsecorrelation (LRC) approach [17], which leads to a differentspatial distribution of ESWL for different response. In thisstudy, the gust loading envelope (GLE) approach [4] is adopted.According to this approach, the BESWL is given as thegust loading envelope scaled by a background factor. Thebackground factor for a given response is defined as theratio of the background response to the response under thegust loading envelope and represents a reduction effect ofpartially correlated wind loads on the background response.The background factors for the displacements at different floorlevels of both buildings are almost constant as 0.83. The shearforce and bending moment at the same floor level have almostsame background factor, which varies over the building heightfollowing an almost linear function from 0.82 at the baseto 0.97 at the top for both buildings. The background factoris insensitive to individual response, which demonstrates theadvantage of the GLE approach for modeling BESWL.

The RESWL is given as the fundamental mode inertialload by distributing the resonant base bending moment orbase torque with additional amplification factors to reflectthe higher mode contributions to story forces at upper floorlevels. The higher mode contributions can also be modeledmore conveniently by introducing a concentrated force at thebuilding top as adopted in current seismic loading specifications(e.g., Chopra [7]). The background and resonant responses arethen quantified separately and combined by using the square-root-of-sum-of-squares (SRSS) rule for the total dynamicresponse.

4.4. Comparison with ASCE 7-05

The alongwind responses of the 20- and 50-story buildingsbased on the wind loads specified in ASCE 7-05 are computedfor the purpose of comparison. The terrain exposure category C,i.e., open country terrain, is chosen according to the conditionsof the wind tunnel experiments. The wind directionality factor,importance factor and topographic factor defined in ASCE 7-05 are taken as unity. The wall pressure coefficients are 0.8on the windward wall and -0.5 on the leeward wall. Like


Fig. 15. Comparison of the mean responses between ASCE 7-05 and measuredpressure (β = 1.25).

the Australian/New Zealand Standard, ASCE 7-05 uses a 3 sgust envelope distribution multiplied by a dynamic responsefactor (referred to as the gust effect factor in ASCE 7-05) todefine the alongwind ESWL. Generally, 3 s gust envelope isdifferent from the mean load distribution. In order to makea meaningful comparison, the gust effect factor defined inASCE7-05 but excluding the reduction factor 0.925 and thegust factor 1 + 1.7gv Iz̄ (where gv is the peak factor and Iz̄ isthe intensity of turbulence at 60% building height) is regardedas the traditional GRF [21], and compared to that from thepressure measurements. Consequently, the ESWL specified inASCE 7-05 divided by the gust effect factor that excludes thefactors 0.925 and 1 + 1.7gv Iz̄ can be regarded as an equivalentmean load profile. Regardless of this treatment, the ESWLand resulting response by ASCE 7-05 remain unchanged.Fig. 3 shows the equivalent mean force coefficient profiles ofboth buildings by ASCE 7-05 as compared to the measuredpressure data. For the 20- and 50-story buildings, 1 + 1.7gv Iz̄are taken as 1.89 and 1.76, respectively. The comparison ofthe resulting mean responses is shown in Fig. 15, which isexpressed as the ratio of the response by ASCE 7-05 to thatby measured pressures. It is seen that ASCE 7-05 providesslightly conservative mean load and response for the 20-story


(a) 20-story building (UH = 40 m/s). (b) 50-story building (UH = 46.6 m/s).

Fig. 16. Comparison of the peak dynamic and total responses between ASCE 7-05 and measured pressure (β = 1.25).

building, but almost identical ones for the 50-story building.Fig. 9 shows the comparison of the GRFs. It is noted thatthe GRF defined in ASCE 7-05, which is associated with theGRF of the top displacement, is close to that derived from thepressure measurements.

Fig. 16 compares the peak dynamic and total (includingmean) responses, which are expressed as the ratio of theresponse by ASCE 7-05 to that through pressure measurements.The peak dynamic response based on pressure measurementsis given as the RMS response multiplied by the peak factor.The peak factors are chosen as 3.8 and 3.5 for the 20- and50-story buildings, respectively. It is seen that ASCE 7-05underestimates the shear forces and bending moments at upperfloor levels, which is primarily attributed to the fact thatthe single GRF associated with building top displacement isadopted in the ESWL and used for the quantification of allresponses. As discussed earlier, the actual GRFs for buildingstory forces at upper floor levels are much higher than thatfor the top displacement. The differences in story forces at thelower floor levels are primarily attributed to the differences inthe mean loads.

The estimation of the building response at upper floor levelsby ASCE 7-05 can be improved through modifying the ESWLdistribution associated with the peak dynamic response. Insteadof following the load distribution similar to mean load, itcan be defined to follow the fundamental modal inertial forcethrough distributing the base bending moment. Clearly, themodified ESWL results in the same base bending momentas the original ESWL defined in ASCE 7-05, but differentestimation of other responses. Fig. 17 shows the ratio of thepeak dynamic and total response under the modified ESWL tothat with measured pressures. It is observed that the modifiedEWSL leads to improved estimation of the response. Theseresults are consistent with those shown in Figs. 10 and 12. Thatthe response ratios are less than 1.0 for the 50-story buildingconcerning the total response is attributed to the fact that thebased bending moment determined by ASCE 7-05 is less thanthat from the pressure measurements.

4.5. Mode shape correction factors

The modal shape correction factor is defined as ηx,y( f ) =√H2SQx,y ( f )/SMx,y ( f ) for the translational mode, and ηθ =√SQθ ( f )/ST ( f ) for the torsional mode, where SMx,y ( f ) and

ST ( f ) are the PSDs of the external loading in terms of the basebending moment and base torque, and SQx,y ( f ) and SQθ ( f )

are the PSDs of the generalized forces. The empirical formulassuggested by Holmes [11] and Holmes et al. [14] are chosen asan example in this study, which are ηx,y =

√4/(1 + 3β) and

ηθ =√

1/(1 + 2β) where β is the power law exponent of thefundamental mode shape.

Fig. 18 compares the mode shape correction factorsdetermined from the measured pressures with those based onHolmes’ empirical formulas over the reduced frequency rangef B/UH = 0.1–6 in the cases of both 20- and 50-storybuildings. The fundamental mode shape is assumed to followa power law with an exponent β = 1.0, 1.25 and 1.5. It canbe observed that the empirical formulas match well with thecorrection factors based on the measured pressure data. Thefactor 0.7 which is widely used in practice for the mode shapecorrelation associated with the torsional mode is conservative.

5. Concluding remarks

The wind load effects of 20- and 50-story buildings inthree primary directions were analyzed using detailed dynamicpressure data measured in a wind tunnel. The results of thisstudy reconfirmed some of the findings of previous studiesusing simplified loading models, and presented some newresults that helped to better understand and quantify wind-induced response of tall buildings.

This study highlighted the variation of the GRFs associatedwith different responses. The GRFs for the alongwind topdisplacement, base shear force and base bending moment areclose to each other. However, use of a single ESWL as the meanwind load multiplied by the GRF associated with the building


(a) 20-story building (UH = 40 m/s). (b) 50-story building (UH = 46.6 m/s).

Fig. 17. Comparison of the peak dynamic and total responses between modified ESWL and measured pressure (β = 1.25).


Fig. 18. Comparison of the mode shape correction factors.

top displacement or base bending moment led to noticeableunderestimates of the story forces at upper floor levels.Advanced modeling of alongwind ESWL can be achieved byusing the GRF, which varies along building elevation, or byusing physically more meaningful load distributions.

The conventional analysis practice involving only thefundamental mode of vibration offered sufficiently accuratepredictions of the building top displacement, top rotationand base bending moment. However, neglecting the highermode contributions to both background and resonant responsesmay noticeably under- and overestimate other responses.The analysis involving only the fundamental mode did notnecessarily lead to lower response due to negative correlationbetween the background modal responses. It was recommendedto use quasi-static analysis in terms of the response influencefunction for accurately estimating the background response,which is equivalent to including all mode contributions. Whenonly the higher mode contributions to the resonant responsewere neglected, building story forces at higher floor levelswere markedly underestimated. The acrosswind acceleration oftaller buildings was shown to be insensitive to higher mode

contributions as compared to the accelerations in the other twodirections and those of lower buildings. As the significanceof higher mode contributions depends on building dynamicsand wind loading characteristics, the perception that highermode contributions to building acceleration in any direction ofvibration and under any wind condition are negligible may notbe taken as granted.

Modeling the background and resonant ESWLs separatelywas more convenient and physically meaningful especiallywhen the background response is significant. The resultsdemonstrated the advantage of the gust loading envelopeapproach for modeling the background loading. The resonantloading can be modeled as the fundamental modal inertialload with additional amplification factors and/or a concentratedforce at the building top to take into account the contributions ofhigher modes. The background and resonant responses are thenquantified separately and combined by using the SRSS rule forthe total dynamic response.

The adequacy of the alongwind load specified in ASCE 7-05was also examined. As the ESWL in ASCE 7-05 was essentiallydefined as the mean load multiplied by the GRF associated with


the top displacement, building response at upper floor levelswas markedly underestimated. The modified ESWL followingthe fundamental inertial load over the building height cansignificantly improve the accuracy of response prediction.

Finally, using spatiotemporally varying wind loads, theadequacy of the empirical mode shape correction factors used inthe current force balance technique was confirmed for both thetranslational and torsional modes, while the factor 0.7 adoptedfor torsional mode was conservative.

Acknowledgements

The support for this work provided in part by the newfaculty startup funds of the Texas Tech University is gratefullyacknowledged. The writers are also grateful to Professor YokioTamura of Tokyo Polytechnic University in Japan for providingthe valuable wind pressure data used in this study.

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