Along wind response of non-classically damped high-rise building

EACWE 5
Florence, Italy
19 th – 23 rd July 2009
Flying Sphere image © Museo Ideale L. Da Vinci
Along wind response of non-classically damped high-rise building
S. Hračov, J. Náprstek, S. Pospíšil, C. Fischer
Institute of Theoretical and Applied Mechanics, v.v.i., Academy of Science of the Czech Republic,
hracov@itam.cas.cz, naprstek@itam.cas.cz, pospisil@itam.cas.cz, fischerc@itam.cas.cz
Keywords: random wind response, non-proportional damping, tall buildings.
ABSTRACT
Dynamic response of high-rise buildings due to wind excitation plays a significant role in the
assessment of their functionality. The wind load induces very often undesired vibration regarding
occupant discomfort. The wind velocity includes a strong random fluctuating component with respect
to time and space. Therefore the dynamic component of the wind speed can be considered as a
stationary random process in individual points of a structure.
Description of the static as well as dynamic components has been given by many authors, e.g.
Simiu & Scanlan (1996), etc. In particular the problem of structure excitation by fluctuating wind
component has been discussed during previous decades; see e.g. Fischer & Pirner (1987), Dyrbye &
Hansen (1996) and many others. As a consequence of the random character of the wind load, the
structure response dynamic component has a random character as well.
Important role modifying the response character plays a damping and its detailed configuration.
Commonly used assumption of proportional damping is satisfactory in many cases providing easier
mathematical modeling. In many cases, however, this assumption is far from the reality, as for
instance when a structure is equipped with tuned mass dampers (TMD). For example Náprstek &
Pospíšil (2006) have shown that a TMD is a significant source of the non-proportional damping. This
fact being taken into consideration results in an interaction of conventional eigen-forms in the
meaning of the structure itself (on the level of a deterministic problem). Moreover, the random
character of the response generates another source of interaction of eigen-forms.
The contribution of this paper consists in an analysis of the wind response of a high-rise building
equipped with TMDs. The paper gives an extension of previous results, see Náprstek & Pospíšil
(2007), representing a generalization of the problem from line type to two dimensional model of a
structure. Only along wind random vibrations due to the effect of wind pressure on windward side are
assumed. An application of the theoretical method is demonstrated subsequently on an assessment of
a real structure - tall building.
1. MATHEMATICAL MODEL
Tall building discrete mathematical model, see e.g. Figure 1, and its response can be described by
Contact person: S. Hračov, Institute of Theoretical and Applied Mechanics, v.v.i., Academy of Science of the Czech
Republic, Prosecká 76, Prague, Czech Republic, hracov@itam.cas.cz

very well known system of differential equations of the second order :
( t) ( t) ( t) ( )
Mu ⋅ + Cu ⋅ + Ku ⋅ = pt . (1)
Analysis of the random response is based on a transformation of the governing system (1) into
generalized coordinates. Using orthonormal eigen-forms u 0 of the undamped system as a Galerkin set
for a solution of Eq. (1):
n
∑
() t = ⋅f
()
u u t . (2)
j=
1
0 j
j
The above Eq. (1) can be rewritten in a normal form (2n-number of DOFs respected):
( t) = ⋅ ( t) + ( )
F Q F P t ,
2n×
2n
F () t ,Q∈
,
2n
P()
t ∈ , (3)
F
() t
() t
() t
⎡f
⎤
= ⎢
⎢ ⎣f ⎥,
⎥ ⎦
Q
⎡
= ⎢ −
2 −
⎣
0 I
⎤
Θ B ⎥ , P() t = ⎢ ⎥⋅ p() t = H⋅
() t
I
⎦
⎡0⎤
⎣ ⎦
p , (4)
Θ
2 2
diag 1
= [ Θ ,..., Θ ]
2 n , 0 T
B= u ⋅C⋅u 0 , u0 = [ u01,..., u 0n]
. (5)
In Eqs (2-5) we have denoted: f(t) – time functions representing the movement in generalized
coordinates; p(t) - vector representing wind load adjoined to individual components of the vector f(t);
Θ 2 - diagonal matrix containing squares of eigen-values of the undamped system; B - reduced
damping matrix C; u 0j - j-th eigen-form of the undamped system.
In the next step the eigen-values ξ j and eigen-forms q j of the system (3) should be evalueted
according for instance to Fischer (2000). The solution algorithm is based on the same principle as the
subspace iteration in the case of a symmetric real matrix. One iteration cycle consists essentially in
complex inverse vector iteration and orthogonalization of the iteration vectors.
2. ANALYSIS OF RANDOM RESPONSE
Assuming the input and output processes in Eq. (3) in the form with spectral differentials of white
noise type processes dФ(ω):
∞
iωt
*
p() t e dΦ ( ω)
, () t ( ω,
t) d ( ω)
= ∫
−∞
Their correlations are given by Wiener-Kchintschin relation:
∞
F = ∫ F Φ . (6)
−∞
⎧ T ⎫
E⎨dΦ( ω) ⋅ dΦ ( ω′ ) ⎬= dir( ω′
−ω) S p ( ω)
dω⋅ dω′
. (7)
⎩
⎭
E{.} is the mathematical mean operator with respect to the Gaussian probability density function.
*
n×
n
Matrix F ( ω,
t)
∈ 2 2 is a matrix of deterministic functions describing transformation of random
excitation in generalized coordinates to individual components of the random response in time. S p (ω)
is the spectral density matrix of processes p(t). Inserting Eqs (6) into the system (3) and applying the
mathematical mean operator one obtains:
∞
∫
−∞
( * ( ) * ( ))
∞
i t
F ω, t + Q⋅F ω, t ⋅Sp( ω) ⋅ dω= H⋅Sp( ω)
⋅e ω dω. (8)
∫
−∞

Change sequence of the mean operator and the integration is allowed due to the independence of the
spectral differentials dΦ
( ω)
. The formula (8) should be fulfilled in every moment t. Therefore it
holds:
* *
i t
F ( ω, t) + Q⋅ F ( ω,
t)
= H⋅e ω
*
, ( t)
F ω , | = = 0 . (9)
If the eigen-values of matrix Q are unique, the solution of Eq. (9) can be given using Laplace
transform (t→ξ) in the form:
2n
*
−1
k
⎛ 1 1 ⎞
FL
( ωξ , ) = ( Q+ I⋅ ξ)
H = ∑ ZH
⎜ − ⎟. (10)
ξ −iω iω−ξk
⎝ξ −iω ξ −ξk
⎠
k=
1
Matrix Z k ( 2n×2n) is expressed in a form of diadic product:
T
Zk = qkR ( ) ⋅q kL ( ). (11)
The vectors q k(L) and q k(R) are the k-th normalized left and right eigen-forms of matrix Q respectively,
while ξ k is the k-th eigen-value of matrix Q. For subcritically damped eigen-forms the eigen-values
form complex conjugate pairs with negative real parts. Inverse transform of Eq. (10) results in:
2n
* 2 k ωt
ξ
( ,
kt
ω t) = ∑ ZH i
F ( e e )
k=
1
iω−ξ
k
t
0
− . (12)
The first term in Eq. (12) characterizes the stationary response part, while the second one the
influence of the initial transition effect. Due to the negative real part of ξ k , the second term is
approaching to zero with increasing time and can be neglected.
The correlation matrix of the response is derived using definition as follows:
⎧ ∞
∞
⎪ * T T*
{ } ∫ ( ω ) ( ω ) ∫ ( ω ) ( ω , )
⎫
T
⎪
KF
( t1, t2) = E F( t1) F ( t2)
= E⎨ F 1,
t1 dΦ 1 dΦ 2 F 2 t2
⎬=
⎪⎩−∞
−∞
⎪⎭
∞
∫
T
( ) ( )
* *
1 p
2
= F ω, t ⋅S ( ω) ⋅F
ω, t ⋅dω.
−∞
With respect to Eq. (12) the stationary state of Eq. (13) can be expressed in the form:
F
()
( )
∞ 2n
T T T
i t k k p l l
t e ω q ⋅q ⋅H⋅S ω ⋅H ⋅q ⋅q
= ∫ ∑ ⋅
−∞ kl , = 1 i k i l
K ⋅ dω . (14)
( ω−ξ )( − ω−ξ
)
To obtain the dispersion matrix quantification the time t in Eq. (14) should be put to zero. The square
roots of diagonal items of dispersion matrix reveal to be the decisive values characterizing the
response to the random excitation.
The stochastic part of response u(t) is determined by the correlation function:
n
{ } ∑ E ( θ ) ( θ )
( ) E ( ) ( )
{ }
K , , , , , ( ) , , ( )
u PRt1 t2 = u Pt1 u Rt2 = u0i P⋅ fi i t1 fj j t2 ⋅u0j R =
ij=
1
T
0
f 11 1 2
0
= u ( P) ⋅K ( t , t ) ⋅u
( R).
P and R stand for two arbitrary points of the structure. Matrix K f11 (t 1 ,t 2 ) denotes correlations between
the functions f i (t) corresponding to eigen-form u 0i .
(13)
(15)

For the stationary case Eq. (15) can be rewritten as:
T
( ) f11
K PRt , , = u ( P) ⋅K ( t) ⋅u ( R). (16)
0
u
0
3. EXCITATION DUE TO WIND TURBULENCE
The random component of wind load is described by the spectral density S p (ω). The correlation
matrix of the external load K p (t) is formulated by equation:
T
{ τ τ } { ∫ 0 ( ) ( , τ) P ( , ) ( )
A
∫ τ dA
A
}
T
() ( ) ( )
K p t = E p p t + = E u P p P ⋅ dA p R t + u R 0 R
⋅ . (17)
Due to the linear character of integrative operator and the mathematical mean operator one obtains for
Eq. (17):
T
T
{ , , } ( ) ∫∫ ( ) ( , , ) ( )
() ( ) ( τ) ( τ)
∫∫
K t = u P ⋅ E p P ⋅ p R t + ⋅ u R ⋅ dA dA = u P K P R t ⋅ u R ⋅dA dA
p 0 0 P R 0 pu 0 P R
A A
AA
∫∫ u 0( P) K pu ( P, R, t) u T
0 ( R)
dAP dAR
. (18)
A A
= ⋅
After the Fourier transformation of Eq. (18) it results in:
( ω) ( ω)
∫∫
T
p = 0( P) ⋅S pu P, R, ⋅ 0( R)
⋅dAPdAR
A A
H H B B
T
0( zP, xP) S pu( zP, zR, xP, xR, ω ) 0( zR, xR)
dzPdzRdxPdxR
0 0 0 0
S u u =
∫ ∫ ∫ ∫ u u . (19)
= ⋅ ⋅ ⋅
The spectral density S pu (P, R, ω) describes correlation of the wind pressures between points P and R
of the structure. Only the along wind pressure component is considered. The spectral density
S pu (P,R,ω) is defined by:
2 2
( ω) ρ
S PR , , = c vv AA ⋅G ( ω)
⋅ χ χ . (20)
pu D P R P R vv z x
The function G vv (ω) represents Davenport’s spectral density of the wind speed’s fluctuation, see
Davenport (1967). Coefficients χ x resp. χ z are aerodynamic admittances that describe space
correlation of the pressure on acting windward area. After substitutions Eq. (20) becomes:
2 2 2 2
− ( Cx xP− xR ) + ( Cz zP−zR
)
2 2 v10
⋅Ω v10
2π
pu ( PRω)
= cDρ
vv P R AP AR
⋅ K ⋅e
2 4/3
S , , 4
ω(1 +Ω )
ω
. (21)
where Ω = 1200ω/(2πv 10 ); K- the roughness coefficient; C x , C z - decay coefficients in respective
directions; c D - drag coefficient; ρ - air density; v P - mean value of the wind speed in point P; v 10 -
mean value of the wind speed in ten meters above the ground; A p - area corresponding to point P.

4. NUMERICAL ANALYSIS
Figure 1: Prototype of a tall building and its FEM model with TMDs.
The proposed theoretical procedure has been used for calculation of the response of the real 100 m
high-rise building equipped with TMDs, see Figure 1. The TMDs are installed at the top of both
towers and tuned to the first eigen-value of the basic system. The weight of each absorber is 200
tones. The discrete mathematical model of structure is created in CALFEM, which is Matlab toolbox
supporting the finite elements method application. The model consists of 62 nodes (including 2
representing absorbers, each one with 1 degree of freedom). The dampers are modeled as
concentrated masses with Kelvin-Voigt damping terms. The first four eigen-modes of the system
obtained from modal analysis are depicted in the Figure 2. The random wind load is defined by the
Davenport's spectral density for wind velocity. The reference mean value of wind speed at 10 m
height is v 10 = 14 m/s.
Factor of the damping non-proportionality of the system as a whole has been examined as follows:
a set of various damping ratio of the TMDs (dashpot absorber) is used (ζ TMD = 0,04 ÷ 4,5), while
constant structural damping is kept constant (ζ= 0,04). Damping matrix of the basic system is
proportional to the mass matrix. The first eight eigen-values of non-proportionally damped system for
different damping ratio of the TMDs are summarized in the Table 1.
ζ TMD 0,0 0,3 0,7 1,0
-0,026 ± 2,628i -0,138 ± 2,809i -0,088 ± 2,823i -2,565 ; -2,618
ξ 1-8 -0,005 ± 2,788i -0,797 ± 2,754i -2,028 ± 2,090i -3,252 ; -3,330
-0,031 ± 3,138i -0,807 ± 2,820i -2,044 ± 2,087i -0,078 ± 2,825i
-0,053 ± 3,674i -0,152 ± 3,570i -0,109 ± 3,516i -0,093 ± 3,510i
Table 1: The first eight complex eigen-values as function of ζ TMD

ω 1 = 2,63 rad/s
ω 2 = 2,79 rad/s
ω 3 = 3,14 rad/s
ω 4 = 3,67 rad/s
Figure 2: First four eigen-modes of the building with TMDs
For every value of the TMD damping the correlation matrix of the response and root-mean-square
(RMS) of the displacements at different levels have been evaluated. The example of RMS values for
displacements of the points (floors of the towers) is depicted in the Figure 3. The final results are
strongly affected by a number of eigen-forms of undamped system being taken into the consideration.
This applies especially in case of random vibrations, because the influence of stochastic interaction of
the eigen-forms is significant. Nevertheless, the interaction of eigen-forms concerning
non-proportionality of a deterministic system itself cannot be neglected. The influence of the
non-proportionality factor can be observed in the Figure 4. This figure represents the response of the
building respecting first five eigen-forms. In the same time the figure demonstrates the outcome of
the optimization process of TMD parameters.
Left tower
Right tower
Figure 3: The RMS displacement of the both towers along its height
(ζ TMD = 0,24)

Figure 4: The RMS displacement of the top of the right tower as function of ζ TMD
for various number of eigen-modes under consideration ( the first eigen-form : o,
the first two e.: +, the first three e.: , the first four e.: ◊, the first five e.: Δ)
5. CONCLUSIONS
The article presents an outline of an assessment method of a structure subjected to the random wind
load component. The analysis is carried out using generalized coordinates following from the
conventional eigen-forms of an investigated structure. Energy contribution of individual eigen-forms
decreases quickly with an increase of their order. It implies a possibility to limit quite easily a number
of eigen-forms needed. It also leads to a significant reduction of time consumption, when stochastic
analysis should be done. The article highlights the necessity of non-proportional damping to be
respected, when passive or active damping equipment (TMD) is installed into the structure. Also
stochastic interactions of eigen-forms are significant although their influence decay exponentially
with difference of eigen-form numbers.
6. ACKNOWLEDGMENTS
The support of Grants GACR 103/09/0094, 103/07/J060, GAAV A200710902, Grant DFG 476 TSE
113/52/0-1, GAAV A200710805 and research plan AV 0Z20710524 (ITAM) are gratefully
acknowledged.
REFERENCES
Simiu E., Scanlan R.H. (1996). Wind Effects on Structures. Fundamentals and Applications to Design, J. Wiley, NY
Fischer O., Pirner M. (1987). Dynamika kotvenych stozaru, Academia, Praha.
Dyrbye C., Hansen S.O. (1996). Wind loads on structures, Wiley and Sons, Chichester.
Náprstek J., Pospíšil S. (2006). “Along wind random vibrations of a slender structure - modelling by continuous
elements”, Proc. Engineering Mechanics 2006, ITAM ASCR, Prague, 8 pgs, CD ROM
Náprstek J., Pospíšil S. (2007). “Along wind random vibrations of a structure with non-proportional damping”,
Proc. 12th Int. Conf. Wind Engineering , Monash University, Cairns (Australia)
Fischer P. (2000). “Eigensolution of nonclassically damped structures by complex subspace iteration”, Computer
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Davenport A.G. (1967). “The dependence of wind Loads on Meteorological parameters”, Proc. International Research
Seminar, Wind Effects on Buildings and Structures, Toronto Press, Toronto, Canada, 19-83.