Engineering, 2010, 2, 290-298
doi:10.4236/eng.2010.24038 Published Online April 2010 (http://www. SciRP.org/journal/eng)
A More Precise Computation of Along Wind Dynamic
Response Analysis for Tall Buildings Built in Urban Areas
Shu-Xun Chen
Guangxi University, Nanning, Guangxi, China
Received November 12, 2009; revised January 25, 2010; accepted February 4, 2010
Abstract
Modern tall buildings are generally built in urban areas, where value of the terrain roughness length is much
greater than that of the general terrain areas, therefore wind-induced vibrations become more pronounced.
The present formulas of numerical analysis of wind-induced response become less accurate. A more accurate
expression of along-wind load spectrum matrix is proposed. On the basis of the expression, structural analy-
sis formula of along-wind displacement and acceleration response are developed and programmed. The ra-
tionality of these formulas are illustrated in examples.
Keywords: Tall Building, Wind Load Spectrum, Wind-Induced Vibration, Response Analysis, Structural
Analysis
1. Introduction
In modern cities, buildings get taller and more slender.
The effects of wind-induced motions become more pro-
nounced. Accurate prediction of structural response to
wind-induced random vibrations displacement and vibra-
tion acceleration at the design stage is a basis for struc-
tural safety design and serviceability design of modern
tall buildings. The wind-induced response consists of
along-wind response, across-wind response and torsion
response [1,2]. The paper focuses on along-wind re-
sponse analysis firstly.
Several quasi-static methods in wind codes in various
countries and approximate methods based on wind tunnel
studies are used to predict the wind-induced response. As
a building gets taller, the wind-induced random vibra-
tions becomes more violent, these methods become less
reliable. So the wind-induced response of modern tall
buildings should be assessed accurately by using the the-
ory of random vibration. Using the numerical analysis
methods of random vibration response of tall buildings,
people calculate structural response according to wind
load spectra. According to present literatures [2-5] whose
representative is literature [3], considering that the values
of terrain roughness lengths are generally very small, and
the turbulence densities are also very small, square items
of the turbulence component are neglected in expression
The project supported by Dr Chun-Man Chan of HKUST.
formula of the wind load in along-wind direction. But,
for the modern tall buildings built in urban areas, the
values of terrain roughness lengths are much more than
ones for general buildings in general open areas. So that
these items should not be neglected for tall buildings, as
they are the same order values with some items that are
not neglected. Otherwise, calculation results of the tur-
bulence response of the tall buildings will have a large
relative error.
To overcome the above-mentioned error, a more ac-
curate expression of wind load spectrum matrixes is
proposed in this paper. This expression is especially ap-
plicable to tall and other important buildings. Even an
approximate form of the accurate expression is more
precise than the present expression [3]. On the basis of
this expression of wind load spectrum, structural analysis
formulas of wind-induced displacement response and
acceleration response in along-wind direction are devel-
oped and programmed. The expression forms of wind
load spectrum and the analysis formulas proposed in this
paper are especially suitable for using the models of the
structural finite element analysis of tall buildings and
their results of vibration modal analysis. These analysis
results of wind-induced response can be directly used in
member stress analysis, structural deformation analysis
and structural serviceability analysis of tall buildings.
Rationality of the expression forms of wind load spec-
trum and the analysis formulas proposed is shown in
comparison examples.

S. X. CHEN 291
2. Expression of Wind Load Spectrum Matrix
2.1. Wind-Induced Load Vector
Wind around tall building flows in a complex and ran-
dom way, so the wind is a random process. Wind loads
acting on windward walls are mainly caused by the
steady flow and the turbulence flow. The wind speed at
node i of the windward walls can be expressed as:
Vi
(t) = V0 i + Vr i (t) (1)
where V0 i is the mean wind speed at node i , V0 i can be
obtained with the following formulas:
V0 i = V0 (hi /h0)

(2)
here V0 is standard wind speed at a reference height h0 ,
e.g. 10 m ,  = 0.12 ~ 0.30 , hi is height of node i. Vr i (t)
in Equation (1) is the fluctuate wind speed at height hi,
its power spectrum density S r i (  ) can be determined by
fomulas of the present representative literature [3].
Because along-wind drag force is directly proportional
to the square of relative wind speed, wind load acting
vertically on node i of windward walls of a tall building
can be expressed as:
Pu i = c i a i (
 /2) [ Vi
(t) - U i' ] 2
=
c i a i (
 /2) [Vi
(t) 2
- 2 Vi
(t) U i' + U i' 2 ] (3)
in which
ci ----- ci is the drag coefficient describing the ratio of the
wind load per unit area at i node and the velocity pres-
sure (
 /2) [ Vi
(t) -U i' ] 2.
a i ----- area distributed at i node;
 ----- air density
U i' ----- along-wind displacement speed of node i .
In Equation (3), since wind speed Vi
(t) is much
greater than the structural displacement speed U i', the
high order small value U i' 2 can be undoubtedly ne-
glected, then we can get
Pu i ( t ) = c i a i (
 /2) [Vi
(t) 2
- 2 Vi
(t) U i' ] (4)
Let
C a i = ci a
i (
 /2) 2 V
i
(t)
 c
i a i
 V
0 i
(5)
Adding C a i to the structural damping coefficient Cs i ,
a total damping coefficient C i is obtained :
C i = C s i + C a i (6)
It is taken as the i-th diagonal element of damping
matrix C of the structural dynamic equation of the tall
building shown in Equation (7):
K U + C U’ + M U” = P (t) (7)
Then the along-wind load vector of the tall building
can be expressed as:
Pu (t) = {c i a i (
 /2) Vi
(t) 2}
( i = 1, 2, …, N ) (8)
where N is the total number of structural freedom de-
grees. In Equation (8), if i does not correspond to the
along-wind load of the nodes of the windward walls,
then ci is taken as zero.
Place Equation (1) into Equation (8) :
Pu (t) = { c i a i (
 /2) [ V0 i + Vr i (t)] 2 }
= { c i a i (
 /2) [ V0 i
2 + 2 V0 i Vr i (t) + Vr i 2
(t)] }
(i = 1, 2,…, N) (9)
In the present literature, e.g. in page 76 of the present
representative literature [3], Vr i
2
(t) is neglected, then
() u t P = { c i a i (
 /2) [ V0 i
2 + 2 V0 i Vr i (t)] }
( i = 1, 2, …, N ) (10)
The author consider that V
r I
2
(t) should not be ne-
glected for tall buildings in urban areas. The author’s
reason is as follows:
In Equation (1), Vr i (t) is the turbulent flow speed at i
node. It is a random process with a zero mean value. Its
variance is determined by the following equations:
 r i
2 (h i) = E [ V r i 2
(t) ] = Vo i
2 I r i 2 (11)
where I r i is the turbulence density at i node, it can be
obtained from the following equation [3]:
I r i =
 r i
(h i) / Vo i = 1 / ln (h i/r0 ) (12)
where hi is the height of i node, r0 is the roughness
length of building terrain, r0 = 10 - 5 ~ 10 meter is deter-
mined by the terrain type. Because most of the modern
tall buildings are usually built in urban areas, roughness
length r0 of urban area is about 1-10 meter [3], which is a
very great value. Vr i
2
(t) is not much less than V0 i
2
, for
instance, when r0 = 3 m and h i = 22 m, then I r i =
 r i
(h i) /
Vo i = 0.5. In Equation (11),  r i
2 (h i)= E [ V r i 2
(t) ] =
Vo i
2 I r i 2 = 0.25 Vo i 2 . When h i = 22 m, the relationship
curve of I r i
2 and r0 is showm in Figure 1:
It is shown in the The relationship curve in Figure 1
that I r i
2
of tall buildings in urban area of is much more
Figure 1. The relationship curve of I r i
2 and r0.

S. X. CHEN 292
than one of general buildings in open area. Considering
Equation (11), Vr i
2
(t) of Equation (9) can not be ne-
glected, and Equation (10) with Vr i
2
(t) neglected can
only be applied to buildings built in general open terrain.
For modern tall buildings in urban areas, Equation (10)
will cause a large relative error.
2.2. Wind Load Correlation Matrix
Correlation matrix of the wind load vector shown as
Equation (9) is
R P u (  ) = E [ Pu (t) Pu T (t +  ) ] (13)
Its element of row i, column j is:
R Pu i j () = c i c j a i a j (
 /2) 2
E { [ V0 i + Vr i (t)] 2
 [ V0 j + Vr j (t +
 )] 2 }
= c i c j a i a j (
 /2) 2
E [ Vo i
2
Vo j
2
+ 2 Vo i
2 Vo j Vr j (t +
 ) + 2 Vo i Vo j
2 Vr i (t)
+ Vo i
2 Vr j 2 (t +
 ) + Vo j
2 Vr i 2 (t)
+ 4 Vo i Vo j Vr i
(t) Vr j (t +
 )
+ 2Vo i Vr i
(t) Vr j 2 (t +
 ) + 2Vo j Vr i
2 (t) Vr j (t +
 )
+ Vr i
2(t) Vr j 2 (t +
 )] (14)
Because V
r i
(t) and Vr j (t +
 ) are stationary ergodic
normal distributive random processes [2,5,6], then
E [ V
r i
(t)] = E [ V
r j
(t +
 )] = 0 (15)
E [ Vr i
2(t)] =
 r i
2 (16)
E [ Vr j
2 (t +
 )] =  r j
2 (17)
E [ Vr i
(t) V
r j
(t +
 )] = R r i j (  ) (18)
E [ Vr i
2 (t) Vr j (t +
 )] = E [ Vr i
(t) Vr j 2 (t +
 )] = 0 (19)
E [ Vr i
2 (t) Vr j 2 (t +
 ) ] =  r i
2
 r j
2 + 2 R r i j 2
(
 ) (20)
Then Equation (14) can be:
R Pu i j (
 ) = c i c j a i a j (  /2) 2
[ Vo i
2
Vo j
2
+ Vo i
2
 r j
2 + Vo j 2
 r i
2 +
 r i
2
 r j
2
+ 4 Vo i Vo j R r i j (  ) + 2 R r i j
2(
 ) ] (21)
Considering Equation (12), the element of row i, col-
umn j of the correlation matrix of the wind load can be
obtained:
R Pu i j (
 ) = c i c j a i a j (  /2) 2
[ Vo i
2
Vo j
2
 (1 + I r i
2 + I r j 2 + I r i 2 I r j 2)
+ 4 Vo i Vo j R r i j (
 ) + 2R r i j
2(
 ) ] (22)
In the present literature [3], Vr i
2
(t) of wind load is ne-
glected as Equation (10), then the correlation matrix
shown in Equation (13) is
( )[() ( ) ]
T
u u
u
= E t t + 
P
RPP  (23)
The element of of row i, column j of the correlation
matrix of the wind load is
( )
P u i j
R

= c i c j a i a j (  /2) 2
[ Vo i
2
Vo j
2
+ 4 Vo i Vo j R r i j (
 )] (24)
Comparing Equation (24) and Equation (22), their
relative error is a quite large for modern tall buildings
built in urban areas.
2.3. Wind Load Spectrum Matrix
From Equation (22), a spectrum matrix of the wind load
vector can be obtained:
S P u (
 ) = [1 / (2  )] R P u (  )



ej


d  (25)
Its element of row i, column j is:
S P u i j (  ) = [1 / (2  )] R
Pu i j (
 )



ej


d  (26)
Placing Equation (22) into Equation (26), the result
is:
S P u i j (
 ) = c i c j a i a j (  /2) 2 [ Vo i
2
Vo j
2
 ( 1 + I r i
2 + I r j 2 + I r i 2 I r j 2 )
 (  )
+ 4 Vo i Vo j S r i j
(  ) + 2 S r i j (  )  S r i j (  )] (27)
in which the symbol  expresses Duhamel integral,  (  )
is the
 - function:
 (  ) = [1/(2  )]



ej


d
 (28)
and
S r i j (  ) = [1/(2  )] R r i j (  )



ej


d  (29)
S r i j (  ) is a cross-spectrum of turbulent wind velocity,
which can be obtained from literature [2,12].
A good approximate expression of Equation (27) is
S P u i j (
 ) = c i c j a i a j (  /2) 2 [ Vo i
2
Vo j
2
( 1 + I r i
2
+ I r j
2 )
 (  ) + 4 Vo i Vo j S r i j (  )] (30)
In the comparison example in this paper, relative error
of Equation (30) and Equation (27) will be discussed.
If Vr i
2
(t) of Equation (9) is neglected as Equation
(10), then the element of row i, column j of the wind load
spectrum matrix is:
()
P u i j
S

= c
i c j a i a j (
 /2) 2 [ Vo i
2
Vo j
2
 (  ) + 4
Vo i Vo j S r i j (  )] (31)
Comparing with Equation (31), the approximate ex-
pression Equation (30) has more items (I r I
2 +I r j 2),
which is the same order value as the item corresponding

S. X. CHEN 293
E [ Uu 2 ( t ) ] = R u (0) =S u (
 ) d  (36)
Its N diagonal elements are mean square values of the
displac d
They are marked as a vector 
u. It includes the follow-
in
i-thmegree. It is obtained from
the following eq
in which S P u m n (  wEqua (2
In Equatin (37
static translation response. Its i-th element is the square
egree.
It
 (  /2) 2
V o m
2
Vo n
2 /
 u
4 (39)
In 7),  u
2 iector
of the dynamndomb
Its i-th elem
vi
 {[ V2
Vo n
2
( I r m
2 +I r n 2 +I 2 I r n 2 )] /
 u
4
+  ( )  2 V S  )
2 S ( ] d (40)
in which S r m n (  ) is shown as Equation (29).
For cce oion,
tion (27 as i
dom
vi
 { [ Vo m
2
Vo n
2
( I r m
2 + I r n 2 )] /
 4
+  H (  )  2 4 V V( ) d  } (41)
to 4 Vo iVo j S r i j (  ), and should not be neglected. Their
necessity is shown in the following formulas of
wind-induced response analysis.
3. Wind-Induced Response Analysis
Wind-induced structural responses in along-wind direc-
tion include a static part, i.e. an along-wind translation,
and a dynamic part, i.e. an along-wind random vibration.
The along-wind translation is mainly caused by the
steady flow. The along-wind vibration is caused by the
turbulent flow. The along-wind random vibration in-
cludes a quasi-steady background turbulence response to
low frequency component and a narrow-band resonant
response.
3.1. Displacement Response Analysis
According to the analysis method of random vibration
response, the calculation formula of spectrum matrix of
the displacement response vector can be expressed as:
S u (
 ) = Hu (  ) S P u (  ) Hu * (  ) (32)
in which the spectrum matrix of wind-induced load
SPu(  ) is shown as Equation (25), Hu*(  ) is the conju-
gate matrixes of frequency response matrix Hu (  ), and
Hu (
 ) =  u H
u (
 )  u T (33)
In which
 u and Hu (  ) are the natural vibration shape
and the frequency response function of the lower
along-wind vibration model respectively. The relationship
between the vibration shape and mass matrix is
 u
T M
 u
= M
u* = 1. Hu (
 ) can be obtained from the following
formula:
Hu (
 ) = 1 / [( 
u
2 -
 2) + j 2 
u

u
 ] (34)
in which 
u and

u are the natural frequency and damp-
ing of the lowest along-wind vibration model.
The element of row i, column j of the displacement
spectrum matrix shown in Equation (32) is:
S u i j (  )= 
i u

j u

m u

n u
 Hu (  ) 
2 SP u m n (
 )
(35)
m
N


1n
N


1
In which 
i u is the i-th element of vibration shape

u
of the along-wind vibration mode. SP u m n (  ) is shown as
Equation (27).
 Hu (  )  is the absolute value of the
frequency response functions of the along-wind vibration
mode shown in Equation (34).
According to the response analysis theory of random
vibration, the across mean square value matrix of the
displacement response is:



ement response of the N noe freedom degrees.
2
g two parts:

u
2 =
 u
2 +
 u
2 (37)
its i-th element is the mean square value of displacement
response of the freedo d
uation:

u i 2 =


Su i i (  ) d  = 
i u
2
N
N 

m u

n u
m1
 ) d 
n1
  Hu (  )  2 SP u m n ( (38)
) is shon as tion 7).
o ),

u
2 is the square value vector of the



value of the static translation of the i-th freedom d
is
 u i
2
=

i u
2
N
N 

m u

n u c m cn am an
m1n1
Equation (3s the mean square value v
ic ra viration displacement response.
ent is the mean square value of the random
bration displacement of the i-th freedom degree. It is
 u i
2
=

i u
2
m
N

1n
N


1

m u

n u c m cn am an (
 /2) 2
o mr m



Hu  [4 o m Vo nr m n (
+ r m n(  )  S r m n  )  }
onvenienf computat SP u m n (  ) of Equa-
) can be takents approximate form shown in
Equation (30). So the mean square value of the ran
bration displacement of the i-th freedom degree of the
tall building
 u i
2
becomes:
 u i
2
=

i u
2
m
N

1n
N


1

m u

n u c m cn am an (
 /2) 2
u



u o mo n S r m n


S. X. CHEN 294
Considering:
 Hu (  )  2 S r m n (  ) d    Hu (0)  2
 S ) 
2S
 rm n   /  u+S (  )[ 
/(2
 
3)] (42)
w1 / (

u rmn
is cross correlation. By convention of the present repre-
sen
r
of the i-th freedom
degree. It is shl
 Vo m
2 Vo n 2 ( Ir m 2 +Ir n 2 ) /
 u
4
+ 
i u
 
c cn am an (
 /2) 2
Vo n
 H2
2
 Vo m Vo n( Ir m+ Ir n
2 + 4 rm n Ir m Ir n ) /
 u
4
(44)
In Equation 3), 2s corresponding t the
dynamhe
i-th freedom degree. It is shown as:
o n
 
u m o m Vo n
 S r m n ( 
u )[

/ (2

u

u
3 )]
If the quasi-static turbulent ground displacement re-
sponse ted according to the
unreasonable approximate Equation (31) as the pr
re


 


r m n (
 ) d  +  Hu (  r m n ( 
) d





u
u
4
r mr nr m n u u u
here
 Hu ( 
u)
 2 = 2 
u

u
2 ) 2,
  =  
u
tative literature [3],  u i
2
of Equation (41) can be di-
vided to the following two pats:
 u i
2
  u b i
2 +
 u r i
2 (43)
in which
 u b i
2 is corresponding to the quasi-static tur-
bulent ground displacement response
own as folows:
 u b i
2
=

i u
2
m
N

1n
N


1

m u

n u cm cn am an (
 /2) 2
2
N

m1n1
m u n u m
N

 4 Vo mu (0)  S r m n (  ) d 



=

i u
2

m u

n u cm cn am an
(
 /2)
m
N

1
22
n
N


1
2
(4 u r i io
ic turbulent resonant displacement response of t
 u r i
2 =

i u
2
m
N

1n
N


1

m u

n u cm cn am an
 2
 Vo mV Hu (  )  2 S r m n (  ) d




u
u
= 
i u
2

m u

n ccn am an
 2 V
m
N
1n
N


1
(45)
of Equation (44) is calcula
esent
presentative literature [3], then it is

u b i
2
=

i u
2
m
N

1n
N


1

m u

n u cm cn am an
(
 /2) 2
 4 Vo m  Hu (0)  2 S r m n (  ) d 
Vo n
n u
2
 Vo mVo n( 4 rm n Ir m Ir n ) /(  u
4 )
6)
In Eqo 44mparing the relative values of the
two iteit is obvious
that the items corresponding to ( Ir i
2 +Ir j 2 ) of Equa
(3
ed acceleration response of a tall building
alues. Similar to
ue of accelera-

 H (  )  2 S P u m n (  ) d 

m u cn am an (
 /2) 2



= 
i u
2
N

m u
 cm cn am an
(
 /2)
m
1
2 2
n
N


1
(4
uatin (), Co
ms (4 rm n I r m I r n and I r m
2 + I r n 2),
tion
0) can not be neglected as Equation (31), since the
neglecting will cause a 50% or much higher relative error
for the quasi-static turbulent ground displacement re-
sponse.
3.2. Acceleration Response Analysis
ind-inducW
are random values with zero mean v
quation (37)-(40), the mean square valE
tion response of the i-th freedom degree can be calcu-
lated as:

u” i
2
=

i u
2
N
N 

m u

n u


4
m1n1
u
= 
i u
m1n1

n u c m
2
N

N






z

4
 H u (  ) 
2 { m
2
n
2 2
+ I r n
2 + I r m 2 I r n 2]
 (  ) + 4 Vo m Vo n S r m n (  )
S r m
Consi

2
 
2

4
 H u ( 
+ 2 S r m n (  )  S r m n (  )] d  (48)
Its approximate expression is shown as follows:
Vo Vo [ 1 + I r m
+ 2n (  )  S r m n (  ) } d 
(47)

4H

2
 = 0, can get dering  u (  ) = 0 when
u” i = i u
m n

1
m u

n u c m cn am an ( /2)
2
N

1
N
   ) 2 [ 4 Vo m Vo n S r m n (  )



S. X. CHEN 295

u” i
2

i u
 (49)
nce to Equa ocalculated
as:

u”
is now widely accepted that wind-induced accelera-
tion response has become the standard for the service-
abity an of motiopelildings.
So is of
an important a
4.
=  

m u

n u c m cn am an

2 2
m
N
1n
N
1
 Vo m Vo n  
4
 H u (  )  2 S r m n (  ) d


With referetin (42), it can be
i
2


i u
2

m u

n u c m cn am an

2
m
N
1n
N


1
 Vo m Vo n S r m n ( 
1 ) [
 
u
/( 2

u ) ] (50)
It
lievalution rception in tal bu
Acceleration response analysis of tall buildings
nd practical significance
Example
Structure of a certain tall building is described as a
50-storey of 7-bay by 10-bay framework with 2550 nodes
and 5100 members shown in Figure 2.
Figure 2. 3D model of the example building
A bay width is 4.57 m and a storey height is 3.66 m.
Details of the framework are shown in its plan view in
Figure 3 and elevation views in Figure 4 and 5.
The framework consists of exterior moment frames and
a braced core, which are rectangles with the same centre.
All beams and columns are rigidly connected while the
diagonal braces are simply connected. Two-storey K-
bracing modules are used on both the south and north faces
of the core, as shown in Figure 4. Single-storey knee-
bracing is used in the west and east faces of the core as
shown in Figure 4.
Figure 3. Plan view of the example
Figure 4. A-A view and drag coefficients c i

S. X. CHEN 296
Figure. 5 B-B view Figure 6. a-a view Figure 7. b-b view
The floors between the exterior frames and raced core
are rigidkg /m3,
which isework
are calculated. American AISC standard sections are used
to size the members: Beams are W24  103 shapes, di-
agonals are W14  90 shapes, and columns are also W14
 233 shapes except that the cruciform columns in the core
(See Figure 3) use pairs of two W14  233 shapes ori-
ented perpendicular to each other.
e reference height h = 10 m, the designed wind speed
at
and third frequencies
of natural vibration are given out. They are correspond-
ing the bending vibration on X-direction, the bending
vibration on Y-direction and the twist vibration of the tall
building respectively.
Firstly, through the modal analysis, the lowest fre-
quency and its vibration shape are obtained. They are
corresponding to the bending vibration in along-wind
direction, i.e. X-direction. Its circular frequency is  u =
0.86572 rad/s, its natural frequency is 0.13778 hertz.
In Table 2, the translation values of the bending vibra-
tion shapes on X-direction (along-wind direction) are listed.
The relationship between the vibration shape and mass
matrix are
 u
T M
 u = Mu* = 1. Because the floors be-
tween the exterior frames and braced core are rigid, the
translation values at all nodes of the same storey are the
same.
Table 1. Natural frequencies
mode
number
circular
frequency
frequency
(hertz )
period
(sec)
The parameters about wind load are given as follows:
th 0
the reference height V h0 = 30.0 m /s 2, the roughness
length of building terrain r0 = 2 m, for Equation (11)
and Equation (12), I r i =  r i
(h i) /Vo i > 17.4% at nodes
over 22 meters. Non-dimensional decay constants Cy =
C z = 10.0 which is employed when cross-spectrums of
turbulent wind are calculated, The drag coefficients c i of
surface nodes in along-wind direction are shown out as
Figure 4.
In Table 1, the first, the second
b
. Mass density of the tall building is 150
employed when node masses of this fram
1 8.6572E-01 8.6572E-01 7.2577E+00
2 9.6824E-01 1.5410E-01 6.4893E+00
3 1.2926E+00 2.0573E-01 4.8608E+00
Table 2. The bending vibration shapes in along-wind direction
Storey X-translation Storey X-translation Storey X-translation Storey X-translation StoreyX-translation
1 .14613E-05 11 .44215E-04 21 .11164E-03 31 .18569E-03 41 .25427E-03
2 .40615E-05 12 .50177E-04 22 .11898E-03 32 .19293E-03 42 .26052E-03
3 .70606E-05 13 .56396E-04 23 .12639E-03 33 .20013E-03 43 .26670E-03
4 .10390E-04 14 .62780E-04 24 .13381E-03 34 .20722E-03 44 .27271E-03
5 .14206E-04 15 .69378E-04 25 .14127E-03 35 .21426E-03 45 .27865E-03
6 .18365E-04 16 .76104E-04 26 .14871E-03 36 .22117E-03 46 .28442E-03
7 .22932E-04 17 .83006E-04 27 .15618E-03 37 .22803E-03 47 .29012E-03
8 .27790E-04 18 .90002E-04 28 .16360E-03 38 .23473E-03 48 .29567E-03
9 .33000E-04 19 .97138E-04 29 .17102E-03 39 .24137E-03 49 .30121E-03
10 .38456E-04 20 .10434E-03 30 .17836E-03 40 .24786E-03 50 .30640E-03

S. X. CHEN 297
espoTable 3. The wind-induced r
Storey  u i 2 (m 2)  u b i2 (m 2)
nse in along-wind direction

u b i2 (m 2)
 u r i 2 (m 2) 
u i2 (m 2)

u” i 2 (m 2/Sec4)
5 245014E-02 .533741E-03 .231407E-03 .571563E-02 .869952E-02 .321051E-02
10 .179546E-01 .391125E-02 .1695
15 .584374E-01 .127301E-01 .5519
20 .132175E+00 .287931E-01 .1248
25
75E
21E
35E
.242297E+00 .527821E-01 .228841E.565224E+00 .860303E+00 .317490E+00
78E-01 .90098.506087E+00
00E-01 8E.197.70
44E-01 .173994E
331E-01 .219907E
49E+00 .265888E.40469
-02 .418841E-01 .637499E-01 .235266E-01
-02 .136322E+00 .207489E+00 .765727E-01
-01 .308335E+00 .469303E+00 .173194E+00
-01
30 .386227E+00 .841360E-01 .3647
35 .557352E+00 .121414E+00 .5264
40 .745866E+00 .162480E+00 .7044
1E+00 .137134E+01
.13001+01 895E+01 3319E+00
+01 .264828E+01 .977335E+00
45 .942683E+00 .205355E+00 .890
50 .113979E+01 .248293E+00 .1076
+01 .334711E+01 .123523E+01
+01 6E+01 .149351E+01
According to Equation (39), Equation (44), Equa-
tion (46), Equation (45), Equ
(50), we respectively calculate
c risln
mean ahe qut ground
displacem  u b i thee vuf
threaproturt
gr diespon
ation (38) and Equation
d the mean square value of
the statitanslation dpaceme t response 
u i
2, the
squ re value of t
ent response
asi-static turbu
2 ,
len
mean squarale o
e unsonable apximate quasi-static bulen
oundsplacement rse

u b i
2
accordi ng e
unreasonable approximate Equation (31), the mean
sq valdynmunan
placement eu r i
2meluee
total displacemenn 
u i
2 an sqre
vaion eons u” i nsieg
one ree loncye
meana Ta l
paring the s of 
2 and
to th
uareue of the aic trbulent resot dis-
rsponse  , the an square va of th
t respo
he accelerat
se
r sp
and the me
e 
2. Co
ua
d rinlue of t
ly thsponse to thwest resonant freque, thes
squ
Com
re values are listed in
value
be 3.
u b i

u b i 2of
Table 2,
th rlati ei reve error is ( 
u b i
2 –

u b i 2) /

u b i
2 = 130.6%.
E  u b ven ( i –

u b i) /

u b i = 51.8 st
supports the above-mentioned
corresponding to (I r i
2 + I r j 2) of Equation (30) can not
be ne as Equ for tags, since
the negting will c or mrelat
error for  u b i
2.
5. Conclusions
For mon tall buin urbth a
ger rouess lengts ofed r
dom viions becompronoe squa
item of turbulence Vr i
2
tion
can notneglected
In thper, a ong-
wind lotrum. Onthis
pressiorm, analla ofed d
placement and acceleration responsend
rection arevelopeammed
As the present usual expression of along-wind load
turbulence component, even
accurate expression proposed
ccan texp n-
ly sup.
en
e Factors,” Journal o
ctur asio1-34.
. Dart, “n Spectrum of Horizontal
inesr th High Winds,” 1963, pp.
211
yrbd S “WiLoac-
.
06.
. Thorton, Ll., Omizll
Structures for Wind Loading ,” Journal of W
ing erod36, 1990, p
nlan, “Wind Effcts on Structrures:
A Introduction to Wind Engineering,” John Wiley &
Sons ork, 198
P. SForceng,” Pss,
Oxford, 1978, pp. 12-88.
A. Pupleace, “Probability
Proucating, 18.
L. ervic Statnd
Load, eel
Con
B. Jal., Damnd
Stiffness in Th
ture of Wring ial
Aerodynamics
[10] Ad Hoc Committee on Serviceability Research, “Struc-
turaty: Araisach
Nee f Strueringo.
12, 6-266
%. This factrongly
conclusion that the items [5] E. Simiu and R. H. Sca
glectedation (31)ll buildin
lecause a 50%uch higher ive [6]
derldings built ian areas wilar-[8]
ghnth, the effec wind-inducan-
brate more
component
unced. Th
(t) of Equa
re
(9) [9]
be .
is pa
ad spec
more accurate expression of al
is proposed the basis of ex-
n foysis formu wind-inducis-
in along-wi
.
di-
e dd and progr
neglects the square item of
an approximate form of the
is more aurate th he present ressio. The exam
ple strong ports this conclusion
5. Referces
[1] A.G. Davnport, “Gust Loadingf the
Strual Divn, 1977, pp. 1
[2] A.G
Gust
venpo
s Nea
The Predictio
e Ground in
194-.
[3] C. D
tures,” John Wiley
ye an. O. Hansen,
& Sons Inc., New York, 1
nd ds on Stru
996, pp
49-1
[4] C. H. Joseph, et a“ptiation of Ta
ind Engi-
neerand Aynamics, Vol. p. 235-244.
Inc., New Y6.
achs, “Wind in Engineeriergamon Pre
[7] , Random Variable and Random
cess”, High Edion Press, Beij983, pp. 18-4
G. Griffis, “S
” Engineering Journal of Am
eability Limit
erican Institute of S
es Under Wi
t
struction, 1993, pp. 23-28.
. Vickery, et
e Reduction of
“The Role of
Wind Effects
ping, Mass a
on Struc-
s,” Journal ind Engineeand Industr
, Vol. 11, 1983, pp. 265-294.
l Serviceabili Critical Appl and Resear
d”, Journal o
1986, pp. 264
ctural Engine
4.
, Vol. 112, N

S. X. CHEN 298
1] D. Surry and E. M. F. Stopar, “Wind Loading of Large Studies at the Laboratory of Building Physics,” KU Leu-
ven, 2002, pp. 36-66.
Data: Comparing
[1
Low Building,” Journal of Civil Engineering, Vol. 16,
1989, pp. 526-542.
[12] F. K. Chang, “Human Response to Motions in Tall Build-
ing,” Journal of Structural Division, Vol. 99, 1973, pp.
1259-1272.
[13] B. Blocken, et al., “Wind, Rain and the Building Envelope:
[14] L. M. Fitzwater and S. R. Winterstein, “Predicting Design
Wind Turbine Loads From Limited
Random Process and Random Peak Models,” Journal of
Solar Energy Engineering, Vol. 123, No. 4, 2001, pp.
364-371.