DSpace at VNU: Wavelet-Galerkin analysis to study the coupled dynamic response of a tall building against transient wind loads

Even though the primary purpose of the study is to examine the feasibility of the proposed analysis method for studying the transient stochastic response of the tall building, a ‘‘frozen

Wavelet-Galerkin analysis to study the coupled dynamic response

of a tall building against transient wind loads

Thai-Hoa Lea,b, Luca Caracogliaa,⇑

a

Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, USA

b

Department of Engineering Mechanics, Vietnam National University, Hanoi, Vietnam

a r t i c l e i n f o

Article history:

Received 20 June 2014

Revised 27 November 2014

Accepted 27 March 2015

Available online 5 August 2015

Keywords:

Wavelet-Galerkin method

Daubechies wavelet

Coupled dynamics

Transient response

Nonstationary wind loading process

Tall buildings

Thunderstorm downburst

a b s t r a c t

The wavelet-Galerkin analysis approach is explored for the solution of the stochastic structural dynamic response of a tall building under transient nonstationary winds The approach is obtained by combining the Galerkin expansion method with basis-functions selected from discrete orthonormal wavelets (namely, the compactly supported Daubechies wavelets) The expansion transforms the stochastic dynamic problem of the tall building, subjected to time-dependent turbulent-induced forces and motion-induced forces, into a system of random algebraic equations in the domain of the wavelet coef-ficients A reduced-order model of a benchmark tall building is employed as a numerical example Nonstationary wind time histories, simulating the loading of a downburst, are artificially generated at discrete points along the vertical axis of the building by using the notions of evolutionary power spectral density of the turbulence and time-dependent amplitude modulation function Important aspects such as the treatment of boundary conditions are examined The paper also aims at investigating the influence of the order of the wavelets and the wavelet resolution on the numerical accuracy of the building response Even though the primary purpose of the study is to examine the feasibility of the proposed analysis method for studying the transient stochastic response of the tall building, a ‘‘frozen’’ thunderstorm downburst model (a first approximation of a slowly-varying time-dependent wind velocity profile with constant wind direction and negligible thunderstorm translation velocity) is also employed Two time-independent synoptic wind velocity profiles (power-law models) and one non-synoptic downburst wind velocity profile (‘‘Vicroy’s model’’) are considered

Ó 2015 Elsevier Ltd All rights reserved

1 Introduction

Tall buildings are sensitive to wind excitation and often

experi-ence large wind-induced vibration due to small structural stiffness,

small structural damping and low fundamental vibration

build-ings result in many engineering design issues, related to both

cumulative damage and ultimate failure Wind-induced stochastic

dynamics of a tall building can potentially involve complex

dynamic problems due to nonlinear structural behavior, motion

assumptions such as linear structural behavior and stationary wind

loads are usually postulated, as a first approximation Also, the

3D-motion coupling is often neglected Therefore, the stochastic response of a building is commonly examined by means of linear elastic reduced-order models, which primarily describe the response features associated with the first fundamental vibration

building usually includes coupled-motion differential equations due to the combination of time-dependent buffeting forces with motion-induced aeroelastic forces Solution to the uncoupled stochastic dynamic response of the building, induced by stationary wind loads, can usually be found in the frequency domain using the Fourier transform, since the stochastic differential equations

the solution of the motion equations in the time domain, needed in the case of coupled nonlinear building response, is not very often pursued since it may lead to complex and computationally

Furthermore, recent investigations have indicated that the fluc-tuating wind processes in extreme and local-convection wind

http://dx.doi.org/10.1016/j.engstruct.2015.03.060

0141-0296/Ó 2015 Elsevier Ltd All rights reserved.

⇑ Corresponding author at: Department of Civil and Environmental Engineering,

Northeastern University, 400 Snell Engineering Center, 360 Huntington Avenue,

Boston, MA 02115, USA Tel.: +1 617 373 5186; fax: +1 617 373 4419.

E-mail address: lucac@coe.neu.edu (L Caracoglia).

Engineering Structures

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / e n g s t r u c t

^

Apðx;zp;tÞ deterministic modulation function

½A11; A½ 12; ½A21; ½A22 WG approximation coefficient matrices

in the x and y coordinates

and y coordinates

y coordinates

deriva-tive

Cohu;pq x;zp;zq

along-wind coherence function between two

Fa;rðz; t; r; _r; €rÞ distributed self-excited force per unit height

Fb;xðz; tÞ; Fb;yðz; tÞ distributed buffeting forces per unit height in

the x and y coordinates

Fa;xðz; t; _x; _yÞ; Fa;yðz; t; _x; _yÞ distributed self-excited forces per unit

height in the x and y coordinates

decomposi-tion of stadecomposi-tionary-turbulence cross spectral density

ma-trix

of rth coordinate

in the x coordinate

in the y coordinate

(sdof system)

wa-velet expansion

Qa;rðt; r; _r; €rÞ generalized motion-induced aeroelastic forces

the x and y coordinates

Qb;xð Þ; Qt b;yð Þ generalized turbulent-induced forces in the x andt

y coordinates

coordinates

turbulence

turbulence

S0u;pp x;zp

along-wind turbulence

S0u;pqx;zp;zq

stationary cross-power spectrum of the ðxÞ along-wind turbulence

Su;ppx;zp;t

evolutionary auto-power spectrum of the ðxÞ along-wind turbulence

Su;pqx;zp;zq;t

evolutionary cross-power spectrum of the ðxÞ along-wind turbulence

Utot;pzp;t

along-wind total wind velocity field at the coordinate

zp

U0pzp;t

upzp;t

; u0

pzp;t along-wind zero-mean stationary and

mpzp;t

; m0

pzp;t cross-wind zero-mean stationary and transient

along-wind ðxÞ displacement, velocity and acceleration, respectively

cross-wind ðyÞ displacement, velocity and acceleration, respectively

coordi-nates xðtÞ; _xðtÞ; €xðtÞ along-wind ðxÞ displacement, velocity and

accel-eration yðtÞ; _yðtÞ; €yðtÞ cross-wind ðyÞ displacement, velocity and

accel-eration

floor) of the building

d0;lk Kronecker delta

X0;0

;½X0;1; ½X0;2 2-term connection coefficient matrices, con-taining respectivelyX0;0lk; X0;1lk; X0;2lk

X0;0lk; X0;1lk; X0;2lk 2-term connection coefficients at the derivative

order 0, at the derivative orders 0 and 1 and at the derivative orders 0 and 2

Xd1 ;d 2

d1; d2

Xd1 ;d2; ;dn

derivative orders d1;d2; ;dn

Other symbols, subscript or superscript indices and operators:

exhibit ‘‘non-synoptic’’ features, in which the wind speed is rapidly

time-varying and velocity fluctuations are no longer stationary but

transient/nonstation-ary winds evolve with time both in amplitude and frequency As

a result, the stochastic response of a tall building due to transient

winds is transient/nonstationary Also, nonlinear effects may no

Therefore, conventional assumptions on stationarity used to

evaluate the stochastic building response can be inadequate in

the case of an extreme local-convection wind because of the

tran-sient amplitude and frequency properties of the wind field These

are usually the main reasons why numerical solutions for the

wind-induced transient stochastic dynamics of a tall building

have attempted to overcome such difficulties, for example by

investigating the characteristics of extreme nonstationary wind

digi-tally simulating the transient/nonstationary wind velocity field

spectrum, a technique extensively used in earthquake engineering,

Since the discovery of the wavelet transform in the late 80s,

wavelets have emerged as a powerful computational tool for

scien-tific analysis Wavelets are calculated as continuously oscillatory

functions and possess attractive features: zero-mean, fast decay,

short life, time-frequency representation, multi-resolution, etc

Therefore, wavelet transforms have been applied to solve various

computational problems in engineering For example, the

continu-ous wavelet transform has been used to generate artificial

nonsta-tionary seismic processes and nonstanonsta-tionary response of simplified

trans-forms have been predominantly employed for signal processing,

identification (e.g.,[24])

Since Daubechies conceived the compactly supported (discrete)

wavelet-based computational tools has rapidly evolved in structural

dynamics The Daubechies wavelets possess the advantageous

prop-erties of being piecewise-defined functions, compact, orthogonal and

of enabling multi-resolution analysis Wavelets can be employed to

represent ‘‘computational solutions’’ at any pre-selected level of

res-olution The latter property makes them particularly useful for

devel-oping an approximating solution to complex problems in structural

dynamics, for example by Galerkin projection approach The

com-bined wavelet-Galerkin analysis method (WG) is a powerful

approach for engineering computations; in this method the

Daubechies wavelets can efficiently be used as a basis of piecewise

functions for the Galerkin projection

The WG method has been successfully applied to several fields

of engineering, such as the solution of partial and ordinary

time-varying parameters of single-degree-of-freedom (sdof)

involves the solution of stochastic structural dynamic problems

of nonlinear structures subjected to transient/nonstationary wind

loading Preliminary investigations on the use of the WG method

for the wind-induced response analysis include the nonlinear

WG method in stochastic dynamics have been observed These

include the accurate treatment of boundary (or initial) conditions, the estimation of the wavelet resolution, arbitrary time duration and computational complexity These aspects have prevented the

WG method from expanding to a wide spectrum of stochastic

bound-ary conditions and wavelet resolution along with an improved computation of the wavelet connection coefficients have been

In consideration of the recent advancements of the WG method

to study stochastic structural dynamic problems, this paper proposes to use the WG method for the simulation of the coupled stochastic response of a tall building due to transient wind loads by reduced-order dynamic models The WG method is employed to transform the time-varying differential equations of motion, which couple the dynamics of the system with the turbulent-induced buffeting forces and the motion-induced aeroelastic forces, into a random algebraic system of equations in the wavelet domain; the unknown wavelet coefficients of this system can be solved very

the verification of the WG method Multivariate evolutionary transient realizations of the wind turbulence field are artificially simulated along the vertical axis of the building, by utilizing the concept of amplitude modulation functions applied to a

to analyze the influence of the order of the Daubechies wavelet and wavelet resolution on the computation of the building response In order to approximately replicate the features of a transient wind, a ‘‘frozen’’ thunderstorm downburst model is employed In this ‘‘frozen’’ thunderstorm downburst model time-independent wind velocity profile, constant wind direction

as well as a fixed downburst center, which neglects the effect of the thunderstorm translation velocity on the downburst loading, are used Two time-independent wind velocity profiles of a synop-tic wind (power-law models) and a non-synopsynop-tic wind profile (‘‘Vicroy’s model’’) are examined

2 Wavelet-Galerkin analysis: background

ated from a ‘‘mother’’ wavelet by scaling (a) and translation (b) parameters, as

wa;bð Þ ¼x 1ffiffiffiffiffiffi

a

j j

a

Wavelets possess very useful properties, which make them particu-larly attractive to represent transient nonstationary signals Wavelets properties include ‘‘dilation’’, ‘‘translation’’ and the con-cept of multi-resolution, which enables a signal to be observed on the simultaneous time-frequency plane Dyadic and compact

con-cept of discrete sampling (k; j are translation and dilation parameters, respectively)

Compactly supported wavelets are functions with non-zero val-ues only within a finite interval and identically zero elsewhere The family of compactly supported Daubechies wavelets ðDÞ is very well suited for engineering computations The ‘‘father’’ scaling

uð Þ ¼x XN1 k¼0

ak2j=2u2jx  k

k¼0ak¼ 2; k; j are translation and dilation parameters; the scaling functions satisfy:

wavelet functions are derived by the father scaling function as

wð Þ ¼x PN2

Rþ1

1ulð ÞxukðxÞdx ¼ d0;lk,

Rþ1

1wlð Þwx kðxÞdx ¼ d0;lk, PN1

k¼0akakþl¼ d0;l, in which l; k are both

of various orders ðNÞ along with their corresponding mother

and D20 (N = 20) The support interval of each pair of scaling

func-tion and wavelet funcfunc-tion widens in the time domain, when the

order of the Daubechies wavelet increases The Wavelet expansion

function and mother wavelet function of order N at a pre-selected

u xð Þ ¼XN x

l¼0

cjlujlð Þ þx Xj

a¼0

XN x

l¼0

jth resolution; cjl¼ hu xð Þ;ujlðxÞi, with the symbol h; i denoting inner

product; cal¼ hu xð Þ;ualðxÞi are ‘‘detailed’’ coefficients at very small

scales a < j; cal¼ hu xð Þ;ualðxÞli is the translation parameter; Nx is

the computational domain If the discrete wavelet decomposition

of uðxÞ at the jth resolution is u xð Þ PN x

l¼0cjlujlð Þ.x Three examples of D4 wavelets on a 100-s interval are indicated

between time and frequency resolution in the wavelet analysis,

i.e., the finer the time resolution the poorer the frequency

resolution and conversely Fortunately, low frequency resolution and high time resolution are usually needed for processing the fundamental (low) frequency components of common signals; high frequency resolution and low time resolution are necessary for higher frequency components

The Galerkin method is a projection method that has been widely applied to the solution of differential equations in struc-tural dynamics and engineering This method seeks for an approximating solution through the projection of the exact solu-tion onto a subspace spanned by a basis of funcsolu-tions The Galerkin projection approximates the exact solution uðxÞ of the equation Au(x) = f by projecting it onto a subspace using a

ul¼ hu;uli ¼Rþ1

1uð Þxulð Þdx and hAux  f ; wi ¼ 0, with ul being

inner-product operation is applied to the original equation in a

l¼1ulhwl;Auli ¼ hwl;f i, one obtains a matrix

Al;k¼ hwl;Auki; fug ¼ ðu1;u2; uN nÞT; ff g ¼ ðhw1;f1i; hw2;f2i;

hwNn;fNniÞT The basis function is often composed of a function, con-taining multiple piecewise sub-functions Each sub-function is pro-jected onto a given interval of the basis function’s domain The weight functions are chosen to be orthogonal to the basis

coef-ficient matrix

In the WG method, the orthonormal and compactly supported Daubechies wavelets can be employed as the basis functions and weight functions in the Galerkin projection to find an approximate

-2

-1

0

1

2

Time (s)

D2 D4 D6 D8 D10 D20

D10

D20 (b)

(a)

-0.5

0

0.5

1

1.5

Time (s)

D2 D4 D6 D8 D10 D20

D2 D4 D6 D8 D10 D20

D10 D20

Fig 1 Daubechies wavelets D2, D4, D6, D8, D10 and D20: (a) father scaling

D4

D4

D4

(a)

(b)

-1 -0.75-0.5 -0.250 0.250.5

-1 -0.75-0.5 -0.250 0.250.5

-1 -0.75-0.5 -0.250 0.250.5 0.751

Time (s)

a=8, b=60

Fig 2 Dilated and translated D4 wavelet: (a) dilation and translation properties,

solution to a time-varying dynamic problem If the time variable is

the resolution j of the wavelet, as:

u tð Þ ¼XN x

l¼1

1u tð ÞuðtÞdt Similarly,

the Daubechies wavelet subspace, as:

_uðtÞ ¼XN x

l¼1

ulu_lðtÞ; uðtÞ ¼€ XN x

l¼1

The derivatives of the wavelets can be obtained correctly in the

limit support, i.e., in the interval [0, N  1] The inner products

between approximating solutions of the displacement, velocity,

are required by the Galerkin expansion Due to the orthogonality

property of the Daubechies wavelets, these are:

uk;XN x

l¼1

ulul

¼XN x

l¼1

uldlk;

uk;XN x

l¼1

ulu_l

¼XN x

l¼1

ulX0;1lk;

uk;XN x

l¼1

ulu€l

¼XN x

l¼1

with

d0;lk¼

Z þ1

1

ukðtÞulðtÞdt;

X0;1lk¼

Z þ1

1

ukðtÞ _ulðtÞdt;

X0;2lk¼

Z þ1

1

where d0;lkis the Kronecker delta andX0;1lk; X0;2lkare 2-term

of appearance’’ l  k is designated by the support (or the order) of

the wavelet

It is noted that the connection coefficients exclusively depend

on the wavelet resolution and the scaling functions within their

limit support but they do not depend on the analytical signal

The 2-term connection coefficients are only necessary for linear

second-order dynamical systems If higher-order derivatives, cross

terms and nonlinear terms in the motion variables exist, 3-term

connection coefficients or even higher multi-term connection

3 Wavelet-Galerkin analysis: computation of connection

coefficients

In a general application of the method, high-order multi-term

connection coefficients can be defined as a function of the scaling

Xd1 ;d 2 ; ;d n

l 1 l 2 l n ðN; jÞ ¼

Z þ1

1

ud 1

l 1ð Þut d 2

l 2ð Þut d n

l nð Þdt ¼t

Z þ1

1

Yn X

i¼1

udi

l idt: ð7Þ

notation d1;d2; ;dndenotes the derivation order (e.g.,ud

l 1¼d

d ul1

dtd );

l1;l2; ;lnare translation indices of the wavelets In most

applica-tions, however, the first and second order of derivation are usually

l 1 l 2 , with d1¼ 0; d2¼ 0; 1; 2 and l1;l2¼ 0; 1; ; N  1 need to be exclu-sively estimated The 2-term connection coefficients can be found

Xd1 d 2

lk ¼

1

ud 1

l ðtÞud 2

kðtÞdx ¼ 2d1 þd 2 1X

p;q

apaq2 lk ð Þþp

1

ud 1ð Þtud 2

qðtÞdx; ð8Þ

are the scaling coefficients of the scaling functions, defined in

The wavelets are compactly supported, therefore the connec-tion coefficients are also defined over a very limited range, depend-ing on the number of supports (or the order of wavelet), indicated

by the index (l  kÞ For instance, the Daubechies wavelet of order N are compactly supported at (N  1) discrete points,

0 6 l; k 6 ðN  1Þ, thus having a total of ð2N  3Þ connection coef-ficients; furthermore, the ð2N  3Þ 2-term connection coefficients can be determined with the indices ðl  kÞ on the support at the discrete ‘‘points’’ ½N þ 2; N þ 3; ; 0; ; N  3; N  2

For facilitating the computations in the WG analysis, a sparse matrix has been used for collecting the compactly supported con-nection coefficients Computation of the concon-nection coefficients of the Daubechies wavelets and accurate treatment of the boundary conditions are also essential to the implementation of the WG analysis The 2-term connection coefficients applicable to an unbounded time interval, derived from the D6 Daubechies wave-let with wavewave-let resolution j = 1 only, were first computed by

inter-val and for resolutions other than j = 1, providing an efficient method for implementation of arbitrary boundary conditions and arbitrary wavelet resolution This study employs the

on the expansion of the original computational domain of the

points to the left of the original computational domain (before the initial time) and (N-1) points to the right of original compu-tational domain (beyond final time) The new compucompu-tational

in the computations

The WG analysis expands a time-varying signal at a pre-selected resolution j Therefore, the initial choice of wavelet resolution ðjÞ is required for the computation of the connection coefficients The resolution parameter of the Daubechies wavelets

scaling function is ‘‘centered’’, given the number of discretization points The wavelet resolution ðjÞ can be approximately found

arbitrary Daubechies wavelets at the arbitrary wavelet resolution

illus-trates some examples of 2-term connection coefficients of the Daubechies wavelet at the orders D4, D6 and D8 and the wavelet resolution j = 6, 8 and for the derivative orders d = {0, 1, 2} For example, the Daubechies scaling function D4 establishes 5 2-term connection coefficients in the support indices ðl  kÞ on

½2; 1; 0; 1; 2; the D6 has 8 connection coefficients according

to the index ðl  kÞ, supported on ½4; 3; 2; 1; 0; 1; 2; 3; 4; the D8 has 13 connection coefficients with ðl  kÞ evaluated on

½6; 5; 4; 3; 2; 1; 0; 1; 2; 3; 4; 5; 6

4 Verification of the Wavelet-Galerkin analysis using a

single-degree-of-freedom dynamical system

The WG analysis is employed to study the response of a simple

oscillator due to random white-noise loading; results are verified

against conventional solution methods, based on numerical

inte-gration The equation of motion is:

where m; c; k are respectively the mass, damping and stiffness

coefficients; f ðtÞ is a random force; initial conditions at t ¼ 0 are

assumed as uð0Þ ¼ 0; _uð0Þ ¼ 0 In the WG analysis, the system

projected into the wavelet domain First, the time-varying

responses and the random force are approximated by using the

equation in the wavelet domain Finally, the sdof motion equation is

obtained in the wavelet domain as:

mXN x

l¼1

X0;2lkulþ cXN x

l¼1

X0;1lkulþ kXN x

l¼1

Eq.(10)must be solved for l = 1, , Nx The resulting algebraic sys-tem can be written in a compact matrix form as:

A

Each element of the matrix [A] becomes Al;k¼ mX0;2lkþ cX0;1lkþ

kd0;lk; it depends on the connection coefficients and it is completely determined by the selected Daubechies scaling function, the

wavelet coefficients of f tð Þ ¼PN x

l¼1flulðtÞ, which are random It is noted that the second-order stochastic dynamic equation has been transformed into a first-order algebraic equation, the solution of which is much simpler and computationally advantageous

compactly supported Daubechies WG analysis

artificially using the Monte Carlo method to illustrate the WG anal-ysis Daubechies scaling function D6 (N = 6) is used The wavelet

sig-nal In this example the wavelet resolution is fitted as j = 6.65, since the time step is set to 0.01 s The connection coefficient matrix [A]

is deterministic and can be pre-calculated

with the following parameters: m = 1 kg, c = 0.0628 Ns/m and

k = 39.4784 N/m, corresponding to a natural frequency of 1 Hz The figure compares the results by WG analysis to the solution

1=4 The error function of the displacement, between the WG anal-ysis results and the ‘‘exact’’ solution by Newmark-b method (NM),

is defined as E %ð Þ ¼ xðNM xWGÞ2=x2

x denotes resultant displacement Error functions and power spectral density functions (PSD) of the response are illustrated

and the PSD is observed between the WG analysis and the Newmark-b method

5 Stochastic dynamic response of tall buildings: mathematical model

5.1 Reduced-order model and equations of motion The reduced-order model of a tall building structure is briefly introduced and described in this section The dynamic equations

of motion are formulated under the assumptions of linear struc-tural response and modal superposition after decomposition into generalized coordinates, which only retain information on the fun-damental modes of the structure, i.e., the first bending modes in the x and y directions of the building The reader is referred to

time-invariant ‘‘frozen’’ direction in the case of the ‘‘frozen’’

-0.1

-0.05

0

0.05

0.1

0.15

Time (s)

Wavelet-Galerkin Newmark-beta (a)

(b)

0

0.5

1

1.5

2

2.5

3x 10

-4

Time (s)

Error of displacement

0 2.5 5 7.5 1012.51517.520

10 -10

10 -5

10 0

Frequency (Hz)

D Wavelet-Galerkin Newton-beta Newmark-beta

Fig 3 Verification of WG solution, examining the response of an sdof system

subjected to white-noise loading, and comparison with numerical integration by

Newmark-b method: (a) displacement, (b) error function and PSD function of the

solution.

Table 1

2-term connection coefficients of Daubechies waveletsXd1 d2

lk ðN; jÞ.

Note: d 1 ;d 2 : derivative indices, l; k: support indices of wavelets, N: wavelet order, j: wavelet resolution

downburst in Section5.3).Fig 4b illustrates the two main lateral

displacement variables and mean-wind and turbulence

compo-nents at a generic elevation z along the vertical axis of the building

The generalized dynamic equation of the rth mode can be written

Mr€rðtÞ þ Cr_rðtÞ þ KrrðtÞ ¼ Qfþ Qbf ;rðtÞ þ Qaf ;rðt; r; _r; €rÞ: ð12Þ

damping and stiffness of the rth mode The variable r ¼ fx; yg is also

used to designate the generalized coordinate of the fundamental

modes: ðxÞ ‘‘along-wind’’ lateral mode in the plane of the mean

wind direction, ðyÞ ‘‘cross-wind’’ transverse mode The quantities

Qf ;r; Qbf ;rð Þ and Qt af ;rðt; r; _r; €rÞ are, respectively, the generalized

mean wind force, the time-dependent generalized buffeting and

motion-dependent loads The system and loading quantities,

Mr¼

Z h

0

U2rð Þm zz ð Þdz;

Kr¼ 4p2n2

rMr;

Qf ;r¼

Z h

0

Urð ÞFz rð Þdz;z

Qbf ;rð Þ ¼t

Z h

0

Urð ÞFz b;rðz; tÞdz;

Qaf ;rðt; r; _r; €rÞ ¼

Z h

0

In the previous equations h is the total height of the building; z is

mode shape function; mðzÞ is the distributed mass of the building per

damping ratios The lateral loading terms are denoted as

Fr; Fb;r; Fa;r; these are, respectively, the distributed mean wind force, the buffeting forces and the self-excited forces per unit height, acting

in the plane (or direction) of the ‘‘rth’’ mode The global response ðRÞ

The distributed lateral wind forces Frð Þ; Fz b;rðz; tÞ; Fa;rðz; t; r; _r; €rÞ

approximation by quasi-steady aerodynamic theory For a

Fxð Þ ¼z 1 2

 

qU zð Þ2DCD;

Fb;xðz; tÞ ¼ 1

2

 

qUðzÞD 2CDu þ C0

D CL

v

;

Fac;xðz;t; _x; _y; hÞ ¼ 1

2

 

qUðzÞD½2CD_x  ðC0

D CLÞ _y þ 1

2

 

qUðzÞ2D½C0

Dh ð14aÞ

Fyð Þ ¼z 1 2

 

qU zð Þ2DCL;

Fb;yðz; tÞ ¼ 1

2

 

qUðzÞD 2CLu þ C0

L CD

v

;

Fa;yðz; t; _x; _y; hÞ ¼ 1

2

 

qUðzÞD½2CL_x  ðC0

L CDÞ _y þ 1

2

 

qUðzÞ2D½C0

time (this hypothesis will later be relaxed in the case of downburst

dimensions of the floor-plan; CD; CL; C0

D; C0

Lare the static force coef-ficients and their derivatives, normalized with respect to D The mean wind load acts along the x direction; the fluctuating

u z; tð Þ; vðz; tÞ The variables h; _x and _y designate torsional rotation, along-wind and cross-wind transverse velocities of the building at

include the effects of vortex shedding, in this first application of the method, even though it has been recognized that vortex shedding effects are relevant to the estimation of the cross-wind response Also, in the subsequent analysis of the response, torsional effects are not considered and the contribution of the angle h is ignored The extension of the numerical model to study the building dynam-ics in the time domain, accounting for such effects, can be readily

It is generally agreed that the numerical solution of the

integration methods and other step-by-step methods is often extremely complex, not very accurate and even impossible for very large systems In many cases modal coupling, influenced by the motion-dependent forces, is neglected during the simulation of the building response in the time domain for the sake of simplicity and to enable the computations

5.2 Stochastic dynamic response of tall buildings in the wavelet domain

combined with the system parameters and generalized forces in

equation of motion of the building response in the global coordi-nates x (along-wind) and y (cross-wind) can be written in the following generalized form:

Mx€xðtÞ þ ½Cx Qxx _xðtÞ  Qyx_yðtÞ þ KxxðtÞ ¼ Qb;xðtÞ; ð15aÞ

My€yðtÞ þ ½Cy Qyy _yðtÞ  Qxy_xðtÞ þ KyyðtÞ ¼ Qb;yðtÞ: ð15bÞ

(a)

(b)

B

(roof top)

x( ,t)

Mean wind

profile

Fb,x( ,t)

u

Horizontal cross section at height

B

D

v

41

20 30

10 5 1

,t ,t

Fig 4 Coordinate system of a rectangular tall building and sectional forces: (a)

In the previous equations Mx; My; Cx; Cy; Kx and Ky are derived

terms associated with the x coordinate, linearly depending on the

generalized force of the y coordinate These quantities can be

Qxx¼ 

Z h

0

/2x½0:5qU zð ÞDCDdz;

Qyx¼ 

Z h

0

/2x½0:5qU zð ÞD C 0D CLÞ

Qxy¼ 

Z h

0

/2y½2 0:5ð ÞqU zð ÞDCLdz;

Qyy¼

Zh

0

/2y 0:5qU zð ÞD C 0L CD

Qb;x¼ 

Z h

0

/xð0:5ÞqU zð ÞD 2CDu þ C0

D CL

v

dz;

Qb;y¼

Z h

0

/y

1

2

 

qUðzÞD 2CLu þ C0

L CD

v

Similar to the WG analysis of the sdof dynamical system, the

time-dependent quantities are approximated by Galerkin projection

obtained by inner product using the orthogonality of the wavelets

(3) resultant algebraic system of equations is numerically solved;

(4) displacement, velocity and acceleration of the stochastic

dynamic response of the structure are estimated The coupled generalized equations of motion in the wavelet domain are (with l ¼ 1; ; NxÞ:

MxXN x

l¼1

X0;2lk þ ðCx QxxÞXN x

l¼1

X0;1lk þ Kx

XN x

l¼1

dlk

xl Qyx

XN x

l¼1

X0;1lk

yl¼ qbxl; ð17aÞ

My

XN x

l¼1

X0;2lk þ ðCy QyyÞXN x

l¼1

X0;1lk þ Ky

XN x

l¼1

dlk

yl fQxy

XN x

l¼1

X0;1lk gxl¼ qbyl: ð17bÞ

the WG-expansion coefficients of the approximate displacements in

WG-based approximate generalized buffeting forces in the x and

y, similar to Eq.(4); X0;1lk ; X0;2lk are 2-term connection coefficients

equations with random coefficients:

(unknown) dynamic displacements The following matrix terms are also defined:

Fig 5 Computational flowchart with the steps of the WG analysis.

½A11 ¼ Mx½X0;2 þ ðCx QxxÞ½X0;1 þ Kx½I;

½A21 ¼ QxyhX0;1i

;

½A22 ¼ MyhX0;2i

þ Cy Qyy

X0;1

in which ½X0;2 and hX0;1i

denote 2-term connection coefficient

solved simultaneously and numerically to find the motion of the

building The WG-based approximating coefficients of the

resultant random velocities and accelerations in the x and y

gener-alized coordinates can be estimated as:

_xl

€

xl

in which f _xlg; f _ylg; f€xlg and f€ylg are the WG-based approximation

coefficients of the velocities and accelerations in x and y

5.3 Simulation of transient wind loads on tall buildings

Multivariate transient wind fields must be digitally simulated at

a series of discrete nodes, located along the vertical axis of the

The digital simulation of a transient wind flow is based

on the theory of evolutionary power spectral density functions

partially-correlated nonstationary processes can be expressed by superposition of partially-correlated stationary processes, modu-lated by a slowly-varying deterministic time function (amplitude modulation) The spectrum of the stationary processes and the deterministic ‘‘modulation function’’ can be selected by matching

a prescribed evolutionary spectrum Therefore, multivariate time-histories of transient wind speed fluctuations can be repro-duced by identifying a suitable deterministic time function to modulate a synthetically-generated sample of a multivariate stationary fluctuating wind process In this study, a realization of the stationary wind speed process is digitally simulated using the spectral representation approach, either based on the Cholesky

of the cross-power spectral density matrix of the turbulence (e.g.,

the two along-wind and cross-wind directions for a tall building

¼ U0z ;t

þ u0z;t

,

-10

-5

0

5

10

Time (s)

(a)

(b)

Fig 6 Effect of modulation function parameters on the digital simulation of

transient wind speed realizations: (a) cosine modulation function; (b) exponential

modulation function.

(a)

(b)

(c)

-10 0 10

Node 41

-10 0 10

Node 30

-10 0 10

Node 20

-10 0

10

Node 10

-10 0 10

Time (s)

Node 5

-10 0

10

Node 41

-10 0

10

Node 30

-10 0 10

Node 20

-10 0

10

Node 10

-10 0 10

Time (s)

Node 5

-10 0

10

Node 41

-10 0

10

Node 30

-10 0 10

Node 20

-10 0

10

Node 10

-10 0 10

Time (s)

Node 5

Fig 7 Digitally-simulated realization of the u-component wind speed fluctuations

at selected nodes for U h ¼ 30 m/s: (a) stationary wind field, (b) transient wind field with cosine modulation function, (c) transient wind field with exponential modulation function.

pzp;t

, in which p is nodal index ðp ¼ 1; 2; ; MÞ; M is the

U0

pzp;t

is now a time-varying ‘‘mean’’ wind velocity

pzp;t and

v0

pzp;t

are the random nonstationary fluctuating components of

pzp;t and

v0

pzp;t

are found by combining two zero-mean stationary random

processes with deterministic frequency-time modulation function,

^

Apðx;zp;tÞ as follows (e.g.,[19]):

u0

pzp;t

¼ ^Apðx;zp;tÞupðzp;tÞ; v0

pzp;t

¼ ^Apðx;zp;tÞvpðzp;tÞ; ð21Þ

where upzp;t

; vpðzp;tÞ are spatially-correlated zero-mean

It must be noted that, in a transient wind such as a downburst, the

time-varying ‘‘mean’’ velocity U0zp;t

is often determined either

hypothesis of time-independent mean wind velocity profile

U0

pzp;t

 U zp

 

 

is the average shape of a ‘‘frozen’’ downburst profile For stationary winds

¼ UðzÞ, i.e., the

slowly-varying mean wind profile is compatible with the

  This hypothesis is, however,

a first approximation of an actual downburst wind

The theory of evolutionary power spectra for transient random fluctuating processes defines the elements of the evolutionary cross spectral matrix of the wind speed fluctuations as:

Sppðx;zp;tÞ ¼ ^Apx;zp;t

2S0ppx;zp

Spqx;zp;zq;t

¼ ^Apðx;zp;tÞ^AT

q ðx;zq;tÞS0pqðx;zp;zqÞ; ð22bÞ

where T and ‘‘⁄’’ denote the transpose and complex conjugate operators The quantities S0ppx;zp

and S0pqx;zp;zq

are the station-ary auto- and cross-power turbulence spectra, respectively, whereas p and q are generic nodal indices The cross-power spec-trum of the stationary turbulence has been empirically estimated

by Davenport spatial coherence function and Harris spectrum (e.g.,[36,47]), as:

S0pqx;zp;zq

¼ Cohpqðx;zp;zqÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

S0ppx;zp

S0qqðx;zqÞ

q

Cohpq x;zp;zq

¼ exp  Cxzp zq

pU z p

þ U z q

!

xS0

ppx;zp 2pr zp 2 ¼ 0:6XðzpÞ

ð2 þ XðzpÞ2Þ5=6

In Eqs.(23a) and (23b)Cohpqðx;zp;zqÞ is the spatial coherence

rðzpÞ ¼ IpUðzpÞ is the standard deviation of the turbulence at zpwith

Ipbeing the turbulence intensity and XðzpÞ ¼ 1600 x

2 p Uðz p Þ The terms UðzpÞ are the time-invariant wind velocities at the building nodes with

‘‘frozen’’ profile of the downburst in the case of nonstationary winds The evolutionary cross spectral matrix of turbulence, written

construc-tion as:

decomposition of the stationary-turbulence cross spectral matrix,

As a result, the multivariate transient process of the along-wind

pzp;t ,

u0

pzp;t

¼ 2 ffiffiffiffiffiffiffiffi

D x

m¼1

Xnx l¼1

j^Apxl;zp;t

jjHpmðxl;zpÞj cos½xlt

 #pmxl;zp;t

Dx¼xup=nx; xupis the upper cut-off circular frequency; nxis the number of circular frequencies, used by the wave-superposition

¼

jH ðx;z Þjei# pm ð x ;tÞ, with # ¼ tan1 Im H ðx;tÞ

=Re H ðx;tÞ

(a)

(b)

(c)

-0.5

0

0.5

x along-wind

-0.1

0

0.1

Time (s)

y cross-wind

-0.5

0

0.5

x along-wind

-0.1

0

0.1

Time (s)

y cross-wind

-0.5

0

0.5

x along-wind

-0.05

0

0.05

Time (s)

y cross-wind

Fig 8 Example of dynamic displacements in the x along-wind and y cross-wind

directions at the rooftop node 41 for mean wind speed U h ¼ 30 m/s: (a) stationary

wind field, (b) transient wind field with cosine modulation function, (c) transient

wind field with exponential modulation function.