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Engineering Structures 29 (2007) 2641–2653

www.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 2007

Available 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 surface

pressures in a wind tunnel. The variations of the gust response factors (GRFs) associated with different alongwind responses are studied, pointing

out the deficiency of the traditional ESWL modeling based on the GRF associated with the top displacement. The higher mode contributions

to 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 with

those based on the wind load specification ASCE 7-05 is also conducted. Furthermore, the practical methodology of modeling ESWLs on tall

buildings involving the influence of higher mode contributions is proposed. Finally, using the spatiotemporally varying wind load information, the

mode shape correction factors required in the high frequency force balance (HFFB) technique are revisited to examine the efficacy of empirical

formulations 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 through

multiple point synchronous scanning of pressures on a building

model surface in a wind tunnel, or by high frequency force

balance (HFFB) measurements, or by simplified wind loading

codes. The synchronous pressure measurements provide a

detailed description of spatiotemporally varying wind loads,

while the HFFB measurements offer an estimate of the

generalized forces of the fundamental modes of vibration. The

wind loading codes give simplified and often conservative

equivalent static wind loads (ESWLs) on isolated buildingswith simple geometric configurations. For tall buildings

with uncoupled mode shapes in primary directions, wind-

induced responses in each primary direction can be analyzed

independently often involving only the fundamental mode.

The fundamental mode contribution dominates global

building 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).

bending moment. However, higher modes may have noticeable

contributions to some responses such as the top acceleration.

The studies by Simiu [19], Kareem [15] and Simiu and

Scanlan [20] using analytical loading models demonstrated that

the contributions from higher modes may reach to about 20%

of the top acceleration. More detailed discussions concerning

the contributions of higher modes to various building responses

using measured spatiotemporally varying wind loads have not

been adequately addressed in the literature.

Current design practice often requires the dynamic wind

loads 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 worldwide

for modeling alongwind loading. This approach leads to an

ESWL given by the mean wind load multiplied by a GRF, often

associated with the top displacement. Studies have shown that

GRF may vary widely for different response (e.g., Holmes [12];

Chen and Kareem [4–6]). Therefore, using a single GRF

for all response components may lead to remarkable over-

or underestimates of building response. Moreover, the GRF

approach falls short in providing physically meaningful ESWLs

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2007 Elsevier Ltd. All rights reserved.

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

for the acrosswind and torsional responses, which are typically

characterized by zero mean wind loading and response. It

should be mentioned that the Australian/New Zealand wind

loading standard AS/NZS 1170.2: [2] has adopted an advanced

ESWL modeling with a dynamic response factor which

increases with increasing height [2]. The advanced modeling

of 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 shape

correction factors for estimating the fundamental generalized

force through measured base bending moment or base torque.

A number of empirical formulas have been suggested in the

literature (e.g., Vickery et al. [23]; Boggs and Peterka [3]; Xu

and Kwok [25]; Ho et al. [10]; Holmes [11]; Holmes et al. [14];

Chen and Kareem [4]). HFFB measurements have also been

used to identify spatiotemporally varying fluctuating wind loads

based on assumed loading models (e.g., Yip and Flay [26]; Xie

and Irwin [24]). The detailed spatiotemporally varying pressuredata provide an opportunity to verify the adequacy of these

empirical formulas used in practice.

This study addresses the wind load effects and ESWLs of tall

buildings based on measured surface pressures of 20- and 50-

story building models. The variations of the GRFs and higher

mode contributions for various building responses are studied in

detail. Furthermore, the comparison of building response with

that 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 technique

are revisited to examine the efficacy of empirical formulas

adopted in current practice.

2. Analysis framework

Consider the wind-excited response of a tall building in

one of three primary directions in which the building exhibits

a one-dimensional uncoupled mode shape of vibration. The

wind load acting at each floor level, denoted by the mean and

dynamic components P̄ j and P j (t ) ( j = 1, 2, . . . , N ; N is the

total number of building floors), are quantified from multiple

point synchronous scanning of pressures on the building model

surface in a wind tunnel. The mean component of a specific

response of interest, R, e.g., displacement, shear force, bending

moment and member force, is estimated by static analysis underthe mean wind load P̄ j and expressed as

¯ R = N

j=1

µ j P̄ j (1)

where µ j is the influence function of response R defined as the

response under a unit load acting at the j th floor.

The dynamic response, R(t ), is determined through modal

analysis:

R(t )

=

N

n=1

Bnqn(t ) (2)

M n[q̈n(t ) + 2ξnωn q̇n(t ) + ω2n qn(t )] = Qn(t ) (3)

where Bn is the modal participation coefficient; qn(t ) is the

generalized displacement; M n, ξn , and ωn are the generalized

mass, damping ratio and circular frequency, respectively;

Qn(t ) =

N j=1 Θ j n P j (t ) is the generalized force; and Θ j n 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 N d modes where the dynamic

effect is significant, and modes N d + 1 to N with natural

frequencies such that their response is essentially quasi-static.

Subsequently, the classical mode displacement superposition

method given by Eq. (2) can be replaced by the static correction

method (e.g., Chopra [7]):

R(t ) = N d n=1

Bnqn(t ) + N

n= N d +1

Bn qnb(t )

= N d n=1

Bnqnr (t ) + N

n=1 Bnqnb(t ) (4)

where qnb(t ) = Qn(t )/K n is the nth background (quasi-static)

generalized displacement and qnr (t ) = qn(t ) − qnb(t ) is the

nth resonant generalized displacement, i.e., the generalized

displacement excluding the quasi-static component; K n = M nω

2n is the nth generalized stiffness.

It is noted that the static correction method essentially

corresponds to the analysis approach of separating the dynamic

response into the background and resonant components. This

approach has traditionally been introduced in the frequency

domain, and widely used in the analysis of wind-induced

response of structures. The quantification of the backgroundresponse including all mode contributions is equivalent to the

quasi-static analysis in terms of the influence function, i.e.,

Rb(t ) = N

j=1

Bn qnb(t ) = N

j=1

µ j P j (t ). (5)

Response analysis can be carried out in either time or

frequency domain. In the time domain scheme, the response

time history is quantified by using the step-by-step integration

method, e.g., Newmark’s numerical method. Based on the

response time history, R(t ), the root-mean-square value (RMS),

σ R , and the peak value, Rmax, can be directly determined.The ensemble-averaged quantities are then computed based on

multiple samples. The peak factor and gust response factor

associated with R(t ) are given as g R = Rmax/σ R and G R =( ¯ R + Rmax)/ ¯ R, respectively.

Alternatively, when the frequency domain scheme is

utilized, the RMS value of response R(t ) is quantified by

combining the background and resonant components as

σ 2 R = σ 2 Rb+ σ 2 Rr

(6)

σ 2 Rb

=

N

j=1

N

k =1

µ jµk σ P j σ Pk r P jk (7)



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

σ 2 Rr =

N d m=1

N d n=1

Bm Bnσ qmr σ qnr r mnr (8)

where σ Rband σ Rr are the RMS values of the background and

resonant response components, respectively; σ P j is the RMS

value of P j (t ); r P jk is the correlation coefficient between P j (t )

and Pk (t ); σ qnr is the RMS value of the resonant generalizeddisplacement in the nth mode and can be estimated using the

closed-form formulation:

σ qnr =1

K n

π f n

4ξn

SQn ( f n) (9)

where SQn ( f ) is the power spectral density (PSD) function

of the generalized force Qn(t ); and r mnr is the correlation

coefficient between the mth and nth resonant modal responses,

which depends on not only the frequency ratio and damping

ratios, but also the coherence of the generalized forces, and can

be estimated as follows [9,5,6]:

r mnr = ρmnr αmnr (10)

ρmnr

= 8√ ξmξn(βmnξm + ξn)β

3/2mn

(1 − β2mn )

2 + 4ξmξnβmn(1 + β2mn ) + 4(ξ2

m + ξ2n )β

2mn

(11)

αmnr = Re[SQmn ( f )]/

SQm ( f )SQn ( f )

f = f m or f n

(12)

where βmn = f m/ f n; and Re denotes the real part of the

complex value.

The accuracy of estimating resonant response through

the closed-form formulations instead of using the numericalintegration of response PSD will be affected by the smoothness

of the PSD of the generalized force around the modal frequency.

The PSD of the generalized force calculated from wind loading

history with limited samples and time duration are often a

jagged function of frequency. To enhance the accuracy of the

predicted resonant response, a “smooth” estimation of the PSD

at the modal frequency can be used in the analysis by averaging

the spectra over the half-power bandwidth.

If the background response is quantified through modal

analysis, similar to the resonant response, the complete

quadratic combination (CQC) should be used:

σ 2 Rb=

N m=1

N n=1

Bm Bnσ qmbσ qnb r mnb (13)

σ qnb = 1

K n

∞0

SQn ( f n)d f = σ Qn

K n(14)

r mnb = r Qmn (15)

where σ qnb and σ Qn are the RMS vales of the nth background

generalized displacement and generalized force; r mnb and r Qmn

are the correlation coefficients between generalized background

displacements and between generalized forces in the mth and

nth modes.

The peak dynamic response including the background and

resonant components is:

Rmax =

g2bσ

2 Rb

+ g2r σ

2 Rr

(16)

where gb and gr are the peak factors for the background and

resonant components, respectively, usually ranging from 3 to 4.

The building acceleration is of interest for the building

habitability design, which can be determined through modal

analysis in either time or frequency domain. When the

frequency domain approach is employed, the RMS values of

the background and resonant components of the nth generalized

acceleration are determined as

σ ̈qnb = 1

K n

∞0

(2π f )4 SQn ( f )d f (17)

σ ̈qnr = ω2nσ qnr = 1

M n

π f n

4ξn

SQn ( f n). (18)

The background acceleration is generally negligibly small as

compared to the resonant component. When only the resonant

component is considered, the RMS value of the acceleration at

the j th floor level, i.e., a j (t ) = N n=1 Θ jn q̈nr (t ), is given by

σ 2a j=

N m=1

N n=1

Θ jm Θ j nσ ̈qmr σ ̈qnr r mnr . (19)

3. Building examples and associated wind loading

3.1. Building dynamic properties

Two shear buildings of 20 and 50 stories with the same

square cross section of 40 m × 40 m and story height of

4 m are chosen as examples. Both buildings are modeled

as 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 primary

direction, thus building response in each direction can be

analyzed independently. The fundamental frequencies in three

primary directions are assumed to be identical, whereas the

torsional frequency is generally larger than the translational

frequency. 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 natural

frequency of steel moment resisting frames suggested in

ASCE 7-05. Four different story stiffness distributions for each

building are considered. In the first three cases, the story

stiffness is assumed to vary over the building height which

is determined based on the prescribed fundamental frequency

and modal shape. The fundamental mode shape is assumed

to follow a power law with an exponent β = 1.0, 1.25, and

1.5, respectively. The last case corresponds to a uniformly-

distributed story stiffness which is determined from the given

fundamental frequency. Fig. 1 shows the mode shapes of the

first 5 modes of the 50-story building. The modal frequencies

of 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 building

with β = 1.25, the frequency ratios are f 2/ f 1 = 2.3 and

f 3/ f 2 = 1.6. The damping ratio for all modes of both buildings

is assumed to be 2%.

3.2. Wind loading

Wind loads on both buildings are derived from multiple

point synchronous scanning of pressures on the building model

surface in a wind tunnel [22]. These experiments were carried

out 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 were

uniformly distributed over the four wall surfaces of the models

of 10 cm × 10 cm × 20 cm and 10 cm × 10 cm × 50 cm for the

20- and 50-story buildings, respectively. Each wall of these twomodels has 10 and 25 layers of pressure taps, respectively. The

wind direction was set normal to the wall face.The sampling interval of wind pressure time history was

0.00128 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 building

top, U H , varies from 20 to 60 m/s, the full-scale duration of

the record ranges from 148 to 50 min for the 20-story building.

To maintain the same mean wind speed profile for the 50-story

building, the range of U H is set from 23.3 to 69.9 m/s and

the corresponding time duration is from 160 to 53 min. Theentire pressure data were divided equally into six independent

samples to compute the structural responses which are then

used for estimating their ensemble-averaged quantities.

The translational forces and torque at each floor level are

determined by integrating wind pressures within the tributary

area. The forces of the stories at which there is no pressure tap

located are determined by interpolating the story forces acting

on two adjacent stories. Fig. 3 shows the mean and RMS story

force coefficients in the three primary directions of the 20- and

50-story buildings. The translational force coefficient is defined

as the ratio of the force to the product of wind speed pressure

at 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 dividing

the building width. It is observed that the mean alongwind

force coefficient increases along the building elevation, whilethe mean acrosswind story force and torque coefficients are

negligibly small. The RMS force coefficients in three primary

directions are almost constant over the building height. Figs. 4

and 5 show the PSDs and RMS values of the generalized forces

associated with the fundamental and second modes in three

directions. The mode shapes are normalized according to unity

generalized masses. It is noted that the acrosswind dynamic

load is greater than the alongwind and torsional loads. The

fundamental modes correspond to larger loads than those of the

higher modes.

4. Results and discussions

4.1. Mean, dynamic responses and GRF

As the buildings with four different story stiffness

distributions show similar characteristics of wind load effects,

only the building with the power law exponent of the

fundamental mode shape β = 1.25 is discussed here. Fig. 6

shows the alongwind displacement, shear force and bending

moment at different building elevations of the 50-story building

at U H = 46.6 m/s calculated from the time history analysis.

The RMS and peak values of the acrosswind and torsional

responses follow similar variations along the building elevation

as the alongwind response. The peak factor of the response

ranges between 3.5 and 4.0. The analysis in the frequency

domain is also carried out which leads to agreeable RMS

response 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 buildings

at varying wind speeds. The building response increases with

wind speed approximately following a power law. The power

law exponent of the background response is 2.0. The resonant

responses of the 50-story building in alongwind, acrosswind

and torsional directions have an exponent of about 2.9, 3.5 and

2.2, respectively. Consequently, the total dynamic responses in

three directions correspond to an exponent of about 2.4, 3.0, and

2.1, respectively. The top accelerations in three directions have

a power law exponent of 2.5–3.4. The acrosswind acceleration

is significantly greater than those in the other two directions.

The power law exponent of resonant response is greater than

2.0, because in addition to the wind load proportional to the

wind speed square, the decrease in the modal reduced frequency

as wind speed increases introduces extra wind load. As the

resonant response increases with wind speed at a faster rate

than the background response, the ratio of the background

to resonant response decreases with increasing wind speed. It

should be noted that the level of modal damping ratio strongly

influences the resonant response and thus the ratio of the

background to resonant component.

The variations of the GRFs associated with different

alongwind responses of both buildings obtained from the time

domain analysis are shown in Fig. 8. The GRFs for the top

displacement, base shear force and base bending moment arealmost same and about 2.0. The GRFs for the shear forces and

bending moments at higher floor levels are remarkably larger.

Moreover, GRFs for the shear force and bending moment at

the same level are almost identical. As pointed in Holmes [12]

and Chen and Kareem [4], the alongwind building response at

higher floor levels will be significantly underestimated using

the ESWL given as the mean wind load multiplied by the

GRF associated with the top displacement or base bending

moment as suggested in most current wind loading codes and

specifications. The variations of GRFs over building height

follow similar features at different wind speeds. As the dynamic

response 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,U H = 46.6 m/s).


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

(a) 20-story building

(U H = 40 m/s).

(b) 50-story building

(U H = 46.6 m/s).

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

response, the GRF slightly increases with increasing wind

speed as shown in Fig. 9.

4.2. Higher modal contributions

Tables 1 and 2 show the influence of higher mode

truncation on the base and top responses of the 50-story

building at U H = 46.6 m/s. Figs. 10 and 11 show the

influence of higher mode truncation on the 50-story building

response expressed as the ratiosσ 2 R1b

+ σ 2 R1r /

σ 2 Rb

+ σ 2 Rr

and

σ 2 Rb

+ σ 2 R1r /

σ 2 Rb

+ σ 2 Rr . These ratios for the 20-story

building are shown in Figs. 12 and 13. The difference between

Figs. 10 and 11, and between Figs. 12 and 13, is in the

estimation of the background response. Figs. 10 and 12 show

the influence of truncating higher mode contributions to both

background and resonant components, while Figs. 11 and 13

are those by truncating higher mode contributions only to the

resonant component.As the background modal response may have noticeable

negative correlation, considering only the fundamental mode

may not necessarily underestimate background response. When

the frequency domain modal analysis procedure is followed,

the CQC scheme should be used for estimating the background

response especially for lower buildings where the background

component is more dominant. In such cases, the analysis of





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

Table 1

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

Top displacement Shear force Bending moment Top acceleration

Top Base Top Base

Alongwind

σ R1b/σ Rb

1.09 2.47 0.76 2.47 0.96 –

σ R1r /σ Rr 1.00 0.81 0.99 0.81 1.00 0.81

σ Rb/σ Rr 0.73 0.26 1.04 0.26 0.83 –

σ 2 R1b

+ σ 2 R1r

/σ 2

Rb+ σ 2

Rr 1.03 1.00 0.87 1.00 0.99 0.81

σ 2 Rb

+ σ 2 R1r

/σ 2

Rb+ σ 2

Rr 1.00 0.82 0.99 0.82 1.00 0.81

Acrosswind

σ R1b/σ Rb

1.16 3.94 0.72 3.94 0.96 –

σ R1r /σ Rr 1.00 0.93 1.00 0.93 1.00 0.92

σ Rb/σ Rr 0.44 0.12 0.70 0.12 0.50 –σ 2

R1b+ σ 2

R1r /σ 2

Rb+ σ 2

Rr 1.03 1.03 0.92 1.03 0.99 0.92

σ 2 Rb

+ σ 2 R1r

/σ 2

Rb+ σ 2

Rr 1.00 0.93 1.00 0.93 1.00 0.92

Table 2

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

Top rotation Torque Top angular acceleration

Top Base

σ R1b/σ Rb

1.15 1.99 0.72 –

σ R1r /σ Rr 0.99 0.64 0.96 0.64

σ Rb/σ Rr 0.39 0.15 0.61 –

σ 2 R1b

+ σ 2 R1r

/σ 2

Rb+ σ 2

Rr 1.01 0.69 0.91 0.64

σ 2 Rb

+ σ 2 R1r

/σ 2

Rb+ σ 2

Rr 0.99 0.65 0.97 0.64

the background response directly using the influence function

is computationally more efficient and thus recommended. For

very tall buildings, the background response becomes less

important and thus can be approximately estimated by the

modal analysis including only the fundamental mode.

In the case of resonant response, as the modal frequencies

of both buildings are well separated, the correlation between

resonant modal responses is negligible. Accordingly, neglecting

the higher mode contributions always results in lower resonant

response. Table 3 summarizes the contributions of first five

modes to the alongwind resonant responses of the 50-story

building, expressed in terms of the ratio of the RMS modal

response to the total RMS response. It is observed that the

higher mode contributions have different influence on different

responses. That is attributed to their different values of modal

participation 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,U H = 46.6 m/s).


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

Table 3

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

Mode number Top displacement Shear force Bending moment Top acceleration

Top Base Top Base

1 1.00 0.81 0.99 0.81 1.00 0.81

2 0.09 0.43 0.15 0.43 0.03 0.43

3 0.02 0.28 0.06 0.28 0.01 0.28

4 0.01 0.18 0.03 0.18 0.00 0.18

5 0.00 0.13 0.02 0.13 0.00 0.13

and base shear force and bending moment, the story forces

at upper floor levels are more influenced by the higher mode

contributions. For the top acceleration of the 50-story building,

considering only the fundamental mode leads to an error of

about 19% in alongwind direction, 8% in acrosswind direction

and 36% in torsion. These results are consistent with those

reported in the literature [19,15,20]. It should be emphasized

that the significance of higher mode contributions depends

on both structural dynamics and wind loading characteristics,

particularly on the PSDs of the generalized forces at the

modal 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,U H = 40 m/s).


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

top acceleration with an error of about 30%, 37% and 39% in

alongwind, acrosswind and torsional directions, respectively.

Figs. 11 and 13 also demonstrate that story forces at upper

floor levels will be significantly underestimated when only

the fundamental mode is involved in the resonant responseand the background response is estimated by the quasi-static

analysis. The influence of higher mode truncation on resonant

component is almost insensitive to the variation of wind speed

as shown in Fig. 14. When both background and resonant

response components are estimated by only considering the

fundamental mode as shown in Figs. 10 and 12, the dynamic

response may be over- or underestimated, particularly for

lower buildings where the background component has more

significant influence on the total dynamic response. It is

noted that the top displacement and base bending moment are

less sensitive to higher mode contributions. In particular, for

buildings with a linear fundamental mode shape, the higher

modes will provide no contribution to the base bending moment

due to the orthogonality between the higher mode shape and the

influence function, although they would affect other response

components.

4.3. Modeling of ESWLs

When only the fundamental mode contribution is consid-

ered, the ESWLs in three primary directions can be defined by

distributing the base bending moment or base torque follow-

ing the fundamental inertial force distribution (e.g. Chen and

Kareem [4]). This concept can also be used in the cases with

consideration of higher mode contributions by further intro-

ducing modification factors as adopted in current seismic load-

ing specifications (e.g., Chopra [7]). However, considering the

distinct 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 and

RESWL) separately, especially when the background response

is noticeable.

The BESWL can be modeled using the load response

correlation (LRC) approach [17], which leads to a different

spatial 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 the

gust loading envelope scaled by a background factor. The

background factor for a given response is defined as the

ratio of the background response to the response under the

gust loading envelope and represents a reduction effect of

partially correlated wind loads on the background response.

The background factors for the displacements at different floor

levels of both buildings are almost constant as 0.83. The shear

force and bending moment at the same floor level have almost

same background factor, which varies over the building height

following an almost linear function from 0.82 at the baseto 0.97 at the top for both buildings. The background factor

is insensitive to individual response, which demonstrates the

advantage of the GLE approach for modeling BESWL.

The RESWL is given as the fundamental mode inertial

load by distributing the resonant base bending moment or

base torque with additional amplification factors to reflect

the higher mode contributions to story forces at upper floor

levels. The higher mode contributions can also be modeled

more conveniently by introducing a concentrated force at the

building top as adopted in current seismic loading specifications

(e.g., Chopra [7]). The background and resonant responses are

then quantified separately and combined by using the square-

root-of-sum-of-squares (SRSS) rule for the total dynamic

response.

4.4. Comparison with ASCE 7-05

The alongwind responses of the 20- and 50-story buildings

based on the wind loads specified in ASCE 7-05 are computed

for the purpose of comparison. The terrain exposure category C,

i.e., open country terrain, is chosen according to the conditions

of 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 0.8

on the windward wall and -0.5 on the leeward wall. Like


Fig. 15. Comparison of the mean responses between ASCE 7-05 and measured

pressure (β = 1.25).

the Australian/New Zealand Standard, ASCE 7-05 uses a 3 s

gust envelope distribution multiplied by a dynamic response

factor (referred to as the gust effect factor in ASCE 7-05) todefine the alongwind ESWL. Generally, 3 s gust envelope is

different from the mean load distribution. In order to make

a meaningful comparison, the gust effect factor defined in

ASCE7-05 but excluding the reduction factor 0.925 and thegust factor 1 + 1.7gv I ̄ z (where gv is the peak factor and I ̄ z is

the intensity of turbulence at 60% building height) is regarded

as the traditional GRF [21], and compared to that from the

pressure measurements. Consequently, the ESWL specified inASCE 7-05 divided by the gust effect factor that excludes the

factors 0.925 and 1 + 1.7gv I ̄ z can be regarded as an equivalent

mean load profile. Regardless of this treatment, the ESWL

and resulting response by ASCE 7-05 remain unchanged.Fig. 3 shows the equivalent mean force coefficient profiles of

both buildings by ASCE 7-05 as compared to the measured

pressure data. For the 20- and 50-story buildings, 1 + 1.7gv I ̄ zare taken as 1.89 and 1.76, respectively. The comparison of the resulting mean responses is shown in Fig. 15, which is

expressed as the ratio of the response by ASCE 7-05 to that

by measured pressures. It is seen that ASCE 7-05 provides

slightly conservative mean load and response for the 20-story




(a) 20-story building (U H = 40 m/s). (b) 50-story building (U H = 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 that

the GRF defined in ASCE 7-05, which is associated with theGRF of the top displacement, is close to that derived from the

pressure measurements.Fig. 16 compares the peak dynamic and total (including

mean) responses, which are expressed as the ratio of the

response 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- and

50-story buildings, respectively. It is seen that ASCE 7-05

underestimates the shear forces and bending moments at upperfloor levels, which is primarily attributed to the fact that

the single GRF associated with building top displacement is

adopted in the ESWL and used for the quantification of allresponses. As discussed earlier, the actual GRFs for building

story forces at upper floor levels are much higher than that

for the top displacement. The differences in story forces at thelower floor levels are primarily attributed to the differences in

the mean loads.The estimation of the building response at upper floor levels

by ASCE 7-05 can be improved through modifying the ESWL

distribution associated with the peak dynamic response. Insteadof following the load distribution similar to mean load, it

can be defined to follow the fundamental modal inertial force

through distributing the base bending moment. Clearly, the

modified ESWL results in the same base bending momentas the original ESWL defined in ASCE 7-05, but different

estimation of other responses. Fig. 17 shows the ratio of the

peak dynamic and total response under the modified ESWL tothat with measured pressures. It is observed that the modified

EWSL leads to improved estimation of the response. These

results are consistent with those shown in Figs. 10 and 12. That

the response ratios are less than 1.0 for the 50-story buildingconcerning the total response is attributed to the fact that the

based bending moment determined by ASCE 7-05 is less than

that from the pressure measurements.

4.5. Mode shape correction factors

The modal shape correction factor is defined as η x, y( f ) = H 2 SQ x , y ( f )/S M x, y ( f ) for the translational mode, and ηθ =SQθ ( f )/ST ( f ) for the torsional mode, where S M x, y ( f ) and

ST ( f ) are the PSDs of the external loading in terms of the base

bending moment and base torque, and SQ x , y ( f ) and SQθ ( f )

are the PSDs of the generalized forces. The empirical formulas

suggested by Holmes [11] and Holmes et al. [14] are chosen as

an example in this study, which are η x, y = √ 4/(1 + 3β) and

ηθ = √ 1/(1 + 2β) where β is the power law exponent of the

fundamental mode shape.Fig. 18 compares the mode shape correction factors

determined from the measured pressures with those based on

Holmes’ empirical formulas over the reduced frequency range

f B/U H = 0.1–6 in the cases of both 20- and 50-story

buildings. The fundamental mode shape is assumed to follow

a power law with an exponent β = 1.0, 1.25 and 1.5. It can

be observed that the empirical formulas match well with the

correction factors based on the measured pressure data. The

factor 0.7 which is widely used in practice for the mode shape

correlation associated with the torsional mode is conservative.

5. Concluding remarks

The wind load effects of 20- and 50-story buildings in

three primary directions were analyzed using detailed dynamic

pressure data measured in a wind tunnel. The results of this

study reconfirmed some of the findings of previous studies

using simplified loading models, and presented some new

results that helped to better understand and quantify wind-

induced response of tall buildings.

This study highlighted the variation of the GRFs associated

with different responses. The GRFs for the alongwind top

displacement, base shear force and base bending moment are

close to each other. However, use of a single ESWL as the mean

wind load multiplied by the GRF associated with the building




(a) 20-story building (U H = 40 m/s). (b) 50-story building (U H = 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 noticeable

underestimates of the story forces at upper floor levels.

Advanced modeling of alongwind ESWL can be achieved by

using the GRF, which varies along building elevation, or by

using physically more meaningful load distributions.

The conventional analysis practice involving only the

fundamental mode of vibration offered sufficiently accurate

predictions of the building top displacement, top rotation

and base bending moment. However, neglecting the highermode contributions to both background and resonant responses

may noticeably under- and overestimate other responses.

The analysis involving only the fundamental mode did not

necessarily lead to lower response due to negative correlation

between the background modal responses. It was recommended

to use quasi-static analysis in terms of the response influence

function for accurately estimating the background response,

which is equivalent to including all mode contributions. When

only the higher mode contributions to the resonant response

were neglected, building story forces at higher floor levels

were markedly underestimated. The acrosswind acceleration of

taller buildings was shown to be insensitive to higher mode

contributions as compared to the accelerations in the other two

directions and those of lower buildings. As the significance

of higher mode contributions depends on building dynamics

and wind loading characteristics, the perception that higher

mode contributions to building acceleration in any direction of

vibration and under any wind condition are negligible may not

be taken as granted.

Modeling the background and resonant ESWLs separately

was more convenient and physically meaningful especiallywhen the background response is significant. The results

demonstrated the advantage of the gust loading envelope

approach for modeling the background loading. The resonant

loading can be modeled as the fundamental modal inertial

load with additional amplification factors and/or a concentrated

force at the building top to take into account the contributions of

higher modes. The background and resonant responses are then

quantified separately and combined by using the SRSS rule for

the total dynamic response.

The adequacy of the alongwind load specified in ASCE 7-05

was also examined. As the ESWL in ASCE 7-05 was essentially

defined as the mean load multiplied by the GRF associated with




the top displacement, building response at upper floor levels

was markedly underestimated. The modified ESWL following

the fundamental inertial load over the building height can

significantly improve the accuracy of response prediction.

Finally, using spatiotemporally varying wind loads, the

adequacy of the empirical mode shape correction factors used in

the current force balance technique was confirmed for both thetranslational and torsional modes, while the factor 0.7 adopted

for torsional mode was conservative.

Acknowledgements

The support for this work provided in part by the new

faculty startup funds of the Texas Tech University is gratefully

acknowledged. The writers are also grateful to Professor Yokio

Tamura of Tokyo Polytechnic University in Japan for providing

the valuable wind pressure data used in this study.

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