1 ABSTRACT This paper examines state-of-the-art analysis and simulation tools for applications to wind engi- neering, introduces improvements recently developed by the authors, and directions for future work. While the scope of application extends to a variety of environmental loads (e.g. ocean waves and earthquake motions), particular reference is made to the analysis and simulation of non-Gaussian features as they appear in wind pressure fluctuations under separated flow regions and non-stationary characteristics of wind velocity fluctuations during a gust front, a thunderstorm or a hurricane. A particular measured non-Gaussian pressure trace is used as a focal point to connect the various related topics herein. Various methods of nonlinear system modeling are first considered. Techniques are then presented for modeling the probability density function of non-Gaussian processes. These include maximizing the entropy functional subject to constraints derived from moment information, Hermite transformation models, and the use of the Kac-Siegert approach based on Volterra kernels. The implications of non-Gaussian local wind loads on the prediction of fatigue damage are examined, as well as new developments concerning gust factor representation of non-Gaussian wind loads. The simulation of non-Gaussian processes is addressed in terms of correlation-distortion methods and application of higher-order spectral analysis. Also included is a discussion of preferred phasing, and concepts for conditional simulation in a non-Gaussian context. The wavelet transform is used to decompose random processes into localized orthogonal basis functions, providing a convenient format for the modeling, analysis, and simulation of nonstationary processes. The work in these areas continues to improve our understanding and modeling of complex phenomena in wind related prob- lems. The presentation here is for introductory purposes and many topics require additional research. It is hoped that introduction of these powerful tools will aid in improving the general understanding of wind effects on structures and will lead to subsequent application in design practice. Analysis and Simulation Tools For Wind Engineering Kurtis Gurley, Michael A. Tognarelli, and Ahsan Kareem Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, IN, 46556
42
Embed
Analysis and Simulation Tools For Wind Engineeringkgurl/Papers/wind.pdf · Analysis and Simulation Tools For Wind ... Civil Engineering and Geological Sciences, University of ...
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
ABSTRACTThis paper examines state-of-the-art analysis and simulation tools for applications to wind engi-
neering, introduces improvements recently developed by the authors, and directions for future work.
While the scope of application extends to a variety of environmental loads (e.g. ocean waves and
earthquake motions), particular reference is made to the analysis and simulation of non-Gaussian
features as they appear in wind pressure fluctuations under separated flow regions and non-stationary
characteristics of wind velocity fluctuations during a gust front, a thunderstorm or a hurricane. A
particular measured non-Gaussian pressure trace is used as a focal point to connect the various related
topics herein. Various methods of nonlinear system modeling are first considered. Techniques are
then presented for modeling the probability density function of non-Gaussian processes. These
include maximizing the entropy functional subject to constraints derived from moment information,
Hermite transformation models, and the use of the Kac-Siegert approach based on Volterra kernels.
The implications of non-Gaussian local wind loads on the prediction of fatigue damage are examined,
as well as new developments concerning gust factor representation of non-Gaussian wind loads. The
simulation of non-Gaussian processes is addressed in terms of correlation-distortion methods and
application of higher-order spectral analysis. Also included is a discussion of preferred phasing, and
concepts for conditional simulation in a non-Gaussian context. The wavelet transform is used to
decompose random processes into localized orthogonal basis functions, providing a convenient
format for the modeling, analysis, and simulation of nonstationary processes. The work in these areas
continues to improve our understanding and modeling of complex phenomena in wind related prob-
lems. The presentation here is for introductory purposes and many topics require additional research.
It is hoped that introduction of these powerful tools will aid in improving the general understanding of
wind effects on structures and will lead to subsequent application in design practice.
Analysis and Simulation Tools For Wind Engineering
Kurtis Gurley, Michael A. Tognarelli, and Ahsan Kareem
Department of Civil Engineering and Geological Sciences,
University of Notre Dame, Notre Dame, IN, 46556
g load
ssues.
tacitly
ed prima-
sses is
essure
ong non-
he non-
xpected
wind
e regions
skewed
ussian
t due to
turbu-
of the
ts in the
faces.
s may
areem et
ed the
a on the
parated
l scales
Letch-
wind
spectral
ctuating
ysis of
hen the
BACKGROUND
Over the last few decades, our understanding of wind-structure interactions and resultin
effects has significantly improved, yet a need remains for further examination of a host of i
Many of the studies encompassing analysis and modeling of wind effects on structures have
assumed that the involved random processes are Gaussian. This assumption has been invok
rily for the convenience in analysis, since information concerning statistics of Gaussian proce
abundant. This assumption is quite valid for loads that involve integral effects of the random pr
field over large areas. Nonetheless, regions of structures under separated flows experience str
Gaussian effects in the pressure distribution characterized by high skewness and kurtosis. T
Gaussian effects in pressure result in non-Gaussian local loads, and give way to increased e
damage in glass panels and higher fatigue effects on other components of cladding.
The probabilistic analysis of pressure fields has been of interest to those involved in
tunnel studies. Peterka and Cermak (1975), and Kareem (1978), demonstrated that in pressur
where the mean pressure was below -0.25, the pressure probability density functions (pdf) are
such that the probabilities for large negative fluctuations are much higher than those for Ga
processes. Similar observations have been also reported by others. It was also noted tha
nonlinear relationships between wind and pressure fluctuations the pdf of pressure under high
lence may be non-Gaussian. Low-rise structures immersed in the highly turbulent lower part
boundary layer, whose structure is further invigorated by the presence of roughness elemen
surroundings, may experience non-Gaussian pressure fluctuations even on their windward
These non-Gaussian effects may be amplified further as the approaching wind fluctuation
depart from a Gaussian process. Similar effects are observed in wave effects on structures (K
al. 1994). Holmes (1981) and Kawai (1983), utilizing quasi-steady and strip theories evaluat
derived pdf of pressure. The resulting distribution showed good agreement with measured dat
surfaces with attached flows. However, as expected, the derived pdf of pressures in the se
regions is not predicted by the quasi-steady theory as the wind-structure interactions at severa
of turbulence may introduce additional components. This observation is again corroborated by
ford et al. (1993) utilizing full-scale data. In an attempt to identify admittance functions for
pressures, Thomas et al. (1995) have noted that the quasi-steady theory fails to model
descriptions of pressures under separated regions despite the inclusion of the square of the flu
velocity term. Similar comments are offered by Tieleman and Hajj (1995) based on their anal
full-scale data. In summary, the quasi-steady theory offers reliable estimates of load effects w
2
e scale
effects
eory is
babi-
eling of
ers and
pressure
ed wind
of basis
titutive
ulation
tput is
loading
n many
r func-
ressure
tening
resent a
s of a
s (e.g.,
). These
ns. For
y
dominant mode of loading is attributed to buffeting, e.g., surface pressures responding to larg
or low frequency turbulence. However, the pressures resulting from wind-structure interaction
cannot be predicted from the quasi-steady theory. A departure from the quasi-steady th
reflected in the non-Gaussian pressure field.
In light of the established inability of quasi-steady theory to predict the dynamics and pro
listic structure of pressure fluctuations in the separated regions, some thoughts on the mod
non-Gaussian processes are presented. This approach holds promise for providing answ
perhaps models for situations in which the quasi-steady theory has failed to do so because
fluctuations are a result of a nonlinear dynamic interaction.
The analysis of nonstationary processes such as transient wind gusts in short, measur
records has been limited due to shortcomings in the Fourier analysis. Here, we apply a set
functions local in both time and frequency to decompose the signal into octave-banded cons
parts. The wavelet transform is useful in the location of energy transfer in time, and in the sim
of nonstationary processes.
MODELING OF NON-GAUSSIAN PROCESSES
In the study of physical systems, the relationship between the input and the system ou
often sought to model the system response. For linear systems, e.g., in the formulation of gust
factors, such a relationship is used for the prediction of extreme response (Davenport, 1964). I
instances in wind engineering, however, the input and output are not related by a linear transfe
tion due to nonlinear characteristics, e.g., the turbulent fluctuations in a hurricane, negative p
fluctuations on building envelopes and associated fatigue of cladding and, in particular, its fas
system. Many approaches are available for modeling nonlinearly related processes. Here we p
brief look at Volterra series systems, as well as several other alternatives.
Volterra Systems
In the Volterra series formulation, the input-output relationship may be expressed in term
hierarchy of linear, quadratic and higher-order transfer functions or impulse response function
Schetzen, 1980, Kareem and Li, 1988, Spanos and Donley, 1991, Kareem and Zhao, 1994
transfer functions can be determined from experimental data or from theoretical consideratio
example, a nonlinear system modeled by Volterra’s stochastic series expansion is described b
, (1)y t( ) h1 τ( )x t τ–( )dτ h2 τ1 τ2,( )x t τ1–( )x t τ2–( )dτ1dτ2 …+∫∫+∫=
3
where and are the first and second-order impulse response functions.
The Fourier transform of the Volterra series expansion up to second order (retaining two terms
on the right hand side) in Eq. 1 gives the response in the frequency domain as
. (2)
For linear systems, the first term on the right hand side of Eq. 2 is all that is needed to describe
the relationship between input and output. This linear model assumes that the Fourier components at
different frequencies are uncoupled. In the first (linear) term on the right hand side of Eq. 2, the
response at frequency is dependent only on input and the transfer function at frequency .
In the case where the system is nonlinear, the Fourier components are coupled, and additional
terms are needed to capture this interaction. The second term on the right hand side of Eq. 2 couples
the response at frequency with pairs of input components at frequencies whose sum or differ-
ence is through the quadratic transfer function (QTF) . Equation 2 describes a system
whose nonlinear component is non-symmetric with respect to the probability density function (e.g. an
even powered polynomial nonlinearity). A third-order system captures the behavior of systems with
both symmetric and non-symmetric nonlinearities (e.g. polynomial nonlinearities with odd and even
powers).
In the case when input and output of a system is available, the information can be
used to estimate the Volterra kernels in Eq. 2 directly. The linear transfer function is given by
. (3)
where is the expected value operator. Here, the numerator is the cross-power spectrum of the
input and output in terms of their Fourier transforms and , and the denomina-
tor is the auto-power spectrum of the input.
Just as is derived from the cross-power spectrum, the QTF is derived from a higher-
order cross-spectrum. The higher-order cross-spectrum between the input and the output
needed to estimate the QTF is called the cross-bispectrum, denoted . Analogous to the
cross-power spectrum in the numerator of Eq. 3, the cross-bispectrum can be expressed in terms of
the expected value of input and output Fourier components as
. (4)
The QTF is given by
h1 τ( ) h2 τ1 τ2,( )
Y fi( ) H1 fi( )X fi( ) H2 f1 f2,( )X f1( )X f2( )f1 f2+ fi=
∑+=
Y fi( ) fi fi
Y fi( ) fifi H2 f1 f2,( )
x n( ) y n( )
H1 fi( )X∗ fi( )Y fi( )⟨ ⟩
X fi( ) 2⟨ ⟩--------------------------------=
⟨ ⟩
x n( ) y n( ) Y f( ) X f( )
H1 fi( )
x n( ) y n( )
Bxxy f1 f2,( )
Bxxy f1 f2,( ) X∗ f1( )X∗ f2( )Y f1 f2+( )⟨ ⟩=
4
. (5)
If no phase coupling exists between and and , then their phases will be ran-
dom and independent, thus the net expected value of the cross-bispectrum will be zero. The formula-
tion for the QTF given in Eq. 5 is valid for a Gaussian input process . The linear and quadratic
transfer functions can also be estimated for a general random input, i.e., without assuming particular
statistics of the input (e.g., Nam, et al., 1990).
Equation 2 addresses a second-order Volterra system, which assumes the nonlinearity is asym-
metric. More generally, higher-order spectral analysis may be applied to nonlinear system identifica-
tion via the higher-order transfer function, which may then be used with a Volterra series similar to
Eq. 2 with additional higher-order terms to model the nonlinear system.
The analogies between the power spectrum and higher-order spectra may be extended to glean
some insight into their physical meaning. The significance of the power spectrum is well
understood to be the decomposition of the signal variance as a function of frequency. Simi-
larly, the bispectrum may be viewed as the decomposition of skewness as a
function of two frequencies, and the trispectrum as the decomposition of kurtosis
as a function of three frequencies. The volumes under the bispectrum and trispectrum yield
the third and fourth central moments, respectively. When viewed in this light, it is apparent that the
existence of higher-order spectra indicates a deviation from Gaussian.
An estimated bispectrum for an experimentally measured wind pressure record is shown in
Fig. 1. This nonzero bispectrum indicates a deviation from Gaussian due to interaction between low
frequency components. For a quadratic nonlinear process that is a square of a narrow-banded linear
process, the bispectrum contains peaks where components of the linear process interact at their sum
and difference frequencies, imparting energy at those frequencies to the resulting nonlinear process.
In this case, the bispectrum does not consist only of sharp peaks, indicating that the pressure record is
not the result of the square of a narrow-banded process, but more likely the output of an at least
partially quadratic system with a wide-banded input. The input process in this case is in fact a wide-
banded wind velocity process. Were it the case that pressure was the result of a cubic nonlinearity
acting on the wind velocity, the bispectrum would not exist, and the trispectrum would reveal the
symmetrically nonlinear relation of the pressure to velocity.
Alt ernatives to Volterra Systems
Several researchers have addressed the modeling of nonlinear systems by means other than a
H2 f1 f2,( ) 12---=
Bxxy f1 f2,( )
X f1( )X f2( ) 2⟨ ⟩-------------------------------------
Y f1 f2+( ) X f1( ) X f2( )
X f( )
Sxx f1( )
x2
t( )⟨ ⟩
Bxxx f1 f2,( ) x3
t( )⟨ ⟩
Txxxx f1 f2 f3, ,( )
x4
t( )⟨ ⟩
5
(1990)
term in
put is
dmit-
s of the
onents
ressure
on the
g the
ut / sin-
cy of
resenta-
r func-
Volterra series expansion and application of higher-order spectra. For example, Bendat
replaced the higher-order frequency domain contribution in Eq. 5 by a zero-memory squared
series with a linear term. Nonlinear pressure on a building from Gaussian wind velocity in
modeled in this fashion to improve upon the modified quasi-steady theory by using multiple a
tance functions (Thomas, et al., 1995). Here, the pressure autospectrum is expressed in term
spectra of the horizontal and vertical fluctuating wind components, the spectra of these comp
squared, and transfer functions in terms of the cross spectra of the inputs with measured p
output. Bendat’s model replaces the second-order Volterra kernel by a linear kernel based
assumption that the QTF is constant along lines normal to the diagonal as in
. (6)
This is equivalent to all frequency pairs for a particular sum or difference frequency containin
same level of phase coupling. This assumption conveniently reduces, for example, a single inp
gle output second-order Volterra model to a two input / single output linear model. The efficien
the analysis is advantageous, and retains limited memory. The error associated with this rep
tion is lumped into a noise or residual spectrum which is minimized with respect to the transfe
0
1
2
3
4
5
0
1
2
3
4
5
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 1. Estimated bispectrum of a measured wind pressure record.
Freq (Hz)Freq (Hz)
H2 f1 f2,( ) A f1 f2+( )=
6
t
tions describing the linear systems in parallel. The nonlinearity is represented, but the assumption of
its form may be restrictive for some systems. The model may be modified to facilitate the input of
non-Gaussian wind velocity (Bendat, 1990).
Neural Networks
Another recently developed approach to nonlinear system modeling is the application of
neural networks. A multi-layered set of processing elements receives input information and uses the
desired final output information to adjust a weighting factor between each of the elements. Figure 2
shows such a network with three weighting layers , where ,
, and and are the number of elements in the and the layers,
respectively. The network in Fig. 2 has two hidden element layers and between the input
and output layers and . In this example, the input layer consists of the input occurring at
the same time as the current output from , and two delayed inputs. then represents the
weighting of the output from the element before its input to element . The output of
each element is a nonlinear function of the weighted linear sum of the output from each of the
elements in the previous layer as in (Kung, 1993)
; (7)
where is a threshold value fixed for each . Various nonlinear functions may be applied a
the elements. One commonly applied function is the sigmoid function
, (8)
where is a parameter to control the shape of .
The element weights in the neural network are adjusted iteratively to minimize the error
between the resulting and desired final output. This is the training phase, in which the optimum model
parameters are identified, where is the number of network layers, and
for the example in Fig. 2 (Kung, 1993).
An example application is shown in Figs. 3 and 4. In Fig. 3, the input is a simulated Gaussian
wind velocity ( ) sampled at 100 Hz for 10 seconds, and the output is the resulting force on a
unit area using . The neural network has two hidden layers of 25 and 30
elements respectively, with a two delay input. The network is given the first second of input / output
data to train itself with, and the figures present the model prediction of later unknown output given
wind velocity input. In Fig. 4, a Gaussian white noise is added to the force output to represent
Wij m( ) m, 1...3= i 1...Nm=
j 1...Nm 1–= Nm Nm 1– mth
mth
1–
ai 1( ) ai 2( )
ai 0( ) ai 3( )
a1 3( ) Wij m( )
aj m 1–( ) ai m( )
bi m( )
bi m( ) Wij m( )aj m 1–( ) θi m( )+
j 1=
Nm 1–
∑= ai m( ) f bi m( )( )= 1 i Nm≤ ≤ 1 m 3≤ ≤
θi m( ) ai m( )
f bi( ) 1
1 ebi σ⁄–
+-----------------------=
σ f bi( )
Wij m( ) m, 1...M= M
M 3=
U u t( )+
F ρCdA U u t( )+( )2 2⁄=
7
se, but
rchitec-
delays,
measurement noise. The neural network does not exactly model the input / output with noi
closely approximates the uncorrupted output. These results will vary as the neural network a
ture is altered, such as varying the number of elements in the layers, changing the number of
etc. (Hertz, et al., 1991, Antsaklis, J.P., 1993).
Figure 2. Multilayer neural network with three weighting layers and two hidden layers(adapted from Kung, 1993).
Wij 1( )
Wij 2( )
Wij 3( )
a1 1( ) a2 1( ) a3 1( ) a4 1( )
a1 0( )
a1 2( )
a1 3( )
a2 2( )
a2 0( )
a3 2( )
a3 0( )
output layer
input layer
hidden layer
hidden layer
uncorrupt outputNN prediction
2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45 2.538
38.5
39
39.5
40
40.5
41
41.5
42
Figure 3. Noiseless force output and neural network model pre-diction.
Time (sec)
8
sential
d pres-
s much
harlier,
d (e.g.,
lterna-
nd and
and
to wind
litera-
ution.
values
not fit
DF of
MODELING OF PROBABILITY DENSITY FUNCTION
The modeling of the probabilistic structure of non-Gaussian pressure fluctuations is es
for a wide range of applications in wind engineering, e.g. accurate determination of design win
sure for glass panels. Large skewness results in probabilities for negative pressure fluctuation
higher than those for Gaussian processes. Series distribution methods, including Gram-C
Edgeworth, and Longuet-Higgins, based on Hermite polynomials, have been commonly use
Ochi, 1986), but tend to exhibit oscillating and negative tail behavior. For extreme response, a
tive means are considered.
The non-Gaussian distribution derived based on the nonlinear relationship between wi
pressure fluctuations with the assumption of Gaussian velocity is valid mostly for windward
leeward faces. This has been shown to fail in separated regions and over surfaces parallel
flow, where quasi-steady theory breaks down. The lognormal distribution has been used in the
ture to model pressure data as the tail of the distribution is higher than for the normal distrib
This often provides values close to the observations, but still fails to predict the occurrence of
far from the mean. Calderone et al. (1994) recently noted that the lognormal distribution does
the pressure data perfectly.
In view of the preceding shortcomings, three alternative approaches to modeling the P
cates the presence of phase coupling among the various component wave frequencies in ma
This coupling results in a non-Gaussian signal which is capable of dramatically altering the re
statistics of a system thought to be subject to Gaussian input. The removal of the coupled phas
mation from the record returns a Gaussian signal with no significant bispectral characteristi
identical autospectra. This demonstrates the potential importance of the phase information in id
ing non-Gaussian signals, and is the basis of the simulation of non-Gaussian signals in the p
section, where the second-order contributions to the linear complex spectral amplitudes re
phase coupling weighted by the QTF. The interpretation of information from the phase spect
non-Gaussian signals is still not well established and is an area of current research in wave me
e.g., Read and Sobey (1987). Higher-order spectral analysis offers a convenient format which h
vided significant insight into phase information, and is currently being used as a tool to identi
tinctive phase characteristics of non-Gaussian wind pressure data.
In a less complicated application of phase tailoring, the injection of constant phase
small frequency range of otherwise random phase results in a signal with characteristics often
in the simulation and analysis of system response to particular types of grouped input. This c
may be applied to simulating concentrated groups of turbulent gusts during a thunderstorm
26
thetic wind records. Further work is needed to identify and quantify the existence of such groups of
gusts in thunderstorms.
Conditional Simulation
Simulation of random velocity and pressure signals at uninstrumented locations of a structure
conditioned on measured records are often needed in wind engineering. For example, malfunctioning
instruments may leave a hole in a data set or information may be lacking due to a limited number of
sensors. This concept is similar to conditional sampling in experiments or numerical simulations. This
field has matured significantly in the last few years (e.g., Borgman, 1990; Vanmarcke and Fenton,
1991; Hoshiya, 1993; Kameda and Morikawa, 1994; Elishakoff et al., 1994). Fundamentally, two
approaches have been introduced in which the simulation is either based on a linear estimation or
kriging, or on a conditional probability density function. Following Borgman’s work on ocean waves
(1990), Murlidharan and Kareem (1993) have developed schemes for conditional simulation of Gaus-
sian wind fields utilizing both frequency and time domain conditioning. The conditional simulation
permits generation of time histories at new locations when one or more time series for the full length
interval are given, and extension of existing records beyond the sampling time for cases where condi-
tioning time series are limited to a small subinterval of the full length. Consider a pair of correlated
Gaussian random vectors and . Let the bivariate normal distribution of these variables be
denoted
, (47)
where is the mean value, and is the auto or cross-covariance between the variables. If a sample
of is measured and denoted as , then it is the conditional simulation of based on the mea-
sured record that is desired. The conditional pdf for given the information on is expressed as
, (48)
and a conditional simulation is provided by
. (49)
Derivations of the covariance matrices and in the time and frequency domains then provide
all that is needed for conditional simulation. Details concerning these matrices for wind simulation
may be found in Murlidharan and Kareem (1993). In a conditionally simulated field, fluctuations at
intermediate points will foll ow the fluctuations of the surrounding locations provided the scale of
V1 V2
p V( ) pV1
V2
Nµ1
µ2
C11 C12
C21 C22
,
= =
µi Cij
V1 v1 V2
V2 V1
p V2 V1 v1=( ) N µ2 C12T
C111–
v1 µ1–( ) C22 C12T
C111–C12–,+( )=
V2 V1 v1=( ) C12T
C111– v1 V1–( ) V2+=
C11 C12
27
ed the
cerns
condi-
s in a
These
ll scale
3 and a
other
lation.
easured
tation 4
from the
ts in a
e target
tion 4.
that the
time
portion
e top
. Both
mpedi-
e been
eveloped
ussian
rocesses,
into the
fluctuations is large. For small scales, the intermediate signal will vary randomly and may exce
fluctuations at surrounding locations. An interesting application of conditional simulation con
generation of wind velocity fluctuation at a large number of grid points as an upwind boundary
tion for a computational study conditioned on measurements at a limited number of location
wind tunnel (Maruyama and Marikawa, 1995).
An example application of Gaussian conditional simulation is shown in Figs. 14 and 17.
examples are based on measured correlated wind velocity records at four elevations on a fu
tower with the mean removed. Figure 14 shows three of these records at stations 1 through
frequency domain conditional simulation of the fourth location based on information from the
three known records. Here, Eq. 49 is used for a uni-dimensional multi-variate conditional simu
Figure 15 is a comparison of the target cospectrum with the cospectrum between the m
records at station 1 and 4, and the cospectrum between the conditionally simulated record at s
and the measured record at station 1. The jaggedness of the cospectra in the figure arises
variance inherent in individual realizations. An ensemble average of many simulations resul
smooth cospectrum which lies along the target cospectrum. Figure 16 is a comparison of th
power spectral density with that from the measured and conditionally simulated records at sta
Figure 17 shows a measured record out to 2500 seconds in the top figure. It is assumed here
record is only available out to 980 seconds, indicated by the darker portion of the signal. A
domain conditional simulation of the record from 980 to 2500 seconds is shown as the lighter
in the bottom figure, based on information from the first 980 seconds. The lighter part of th
figure indicates the portion of this signal which is not known when generating the bottom figure
examples demonstrate the effectiveness and utility of conditional simulation.
In cases involving non-Gaussian processes, the conditional simulation schemes suffer i
ments like their Gaussian counterparts. In Elishakoff et al. (1994), iterative schemes hav
utilized to simulate non-Gaussian processes. The authors sought to combine the techniques d
for unconditional simulation of non-Gaussian processes and the procedure of conditional Ga
processes. The non-Gaussian known processes are mapped into underlying Gaussian p
where conditional simulation is done. These simulated time histories are then mapped back
non-Gaussian domain.
28
l
0 500 1000 1500 2000 250030
40
50
60
70
80
90
100
110
−2
0
2
conditional
1
2
3
4
Figure 14. Measured wind velocity at 40, 60, and 80 meters, frequency domain conditionasimulation at 100 meters based on records at lower three stations
conditional station 4 / measured station 1measured station 4 / measured station 1 target
Power Spectral Densities of Conditionally Simulated data at station 4
Figure 16. Target, measured, and conditionally simulated power spec-tral density at station 4
Figure 17. Measured wind velocity (top figure) and time domain condi-tional simulation (light portion of bottom figure).
conditioning intervalsimulation
0 500 1000 1500 2000 2500
−2
−1
0
1
2
Time domain conditionally simulated process
0 500 1000 1500 2000 2500
−2
−1
0
1
2
Given time series to be simulated
30
WAVEL ET TRANSFORMS
The inability of conventional Fourier analysis to preserve the time dependence and describe
the evolutionary spectral characteristics of nonstationary processes requires tools which allow time
and frequency localization beyond customary Fourier analysis. The short-term Fourier transform
(STFT) provides time and frequency localization to establish a local spectrum for any time instant.
The problem is that high resolution cannot be obtained in both time and frequency domains simulta-
neously. The moving window must be chosen for locating sharp peaks or low frequency features,
because of the inverse relation between window length and the corresponding frequency bandwidth.
This drawback can be alleviated if one has the flexibility to allow the resolution in time and
frequency to vary in the time-frequency plane to reach a multi-resolution representation of the pro-
cess. This is possible if the analysis is viewed as a filter bank consisting of band-pass fil ters with con-
stant relative bandwidths. One type of local transform is the recently developed wavelet transform
(WT) which decomposes a signal using wavelet functions. Fourier methods of signal decomposition
use infinite sines and cosines as basis functions, whereas the wavelet transform uses a set of orthogo-
nal basis functions which are local. Various dilations and translations of a parent wavelet are joined to
form the family of basis functions. This allows the retention of local transient signal characteristics
beyond the capabilities of the harmonic basis functions. The wavelet transform allows a multi-resolu-
tion representation of a process and provides a flexible time-frequency window which narrows to
observe high-frequency energy content, and broadens to capture low frequency phenomena.
Brief Wavelet Overview
Development of the parent wavelet begins with the solution of a dilation equation to determine
a scaling function , dependent on certain restrictions. The scaling function is used to define the
parent wavelet function, . The basis functions used to represent the signal are defined by transla-
tions and dilations of the parent wavelet. The shape of the parent wavelet is not a single unique shape,
but depends on the desired wavelet order.
The signal being decomposed must consist of samples, where is an integer. Wavelet
analysis decomposes the signal into levels, where the level is denoted as , and the levels are
numbered . Each level consists of translated and partially overlapping
wavelets equally spaced intervals apart. The wavelets at level are dilated such that an
individual wavelet spans of that levels intervals, where is the order of the wavelet being
applied. Each of the wavelets at level is scaled by a coefficient determined by the for-
ward wavelet transform, a convolution of the signal with the wavelet. The notation is such that cor-
φ n( )
ψ n( )
2M
M
M 1+ i
i 1 0 1 ...M 1–, , ,–= i j 2i
=
2M
j⁄ j 2i
= i
N 1– N
j 2i
= i ai j,
i
31
n as a
s. The
o per-
ation of
other-
of the
ion. The
domain
rying
well to
ds to a
d and
k. The
ds the
ft block
d-pass
nel or
relative
wn in
er rela-
to first
sonance
ify, e.g.,
ructure,
wind and
either
other
velet
responds to the wavelet dilation, and is the wavelet translation in level . is often writte
vector , where . There are as many wavelet coefficients as signal sample
level is the signal mean value (Newland, 1993). A variety of packages are available t
form discrete wavelet transform (DWT) analysis (e.g. Kareem et al., 1993).
Applications to Wind Engineering
The present research concerns the use of wavelets to aid in the analysis and simul
nonstationary data. Multi-scale decomposition of processes utilizing wavelets reveals events
wise hidden in the original time history. Wavelet coefficients may be used to derive an estimate
power spectrum. These estimates may be extended to multivariate, e.g., cospectral estimat
wavelet coefficients provide the scalogram, which describes the signal energy on a time-scale
over a range of logarithmically spaced frequency bands. This facilitates identification of time-va
energy flux and spectral evolution. The property of accurate energy representation lends itself
signal reconstruction and simulation. A stochastic manipulation of the wavelet coefficients lea
simulation which is statistically similar to the original signal.
Wavelet Filterbank
Figure 18 presents the time history of the response of a large floating structure to win
wave loads, and the resulting band-passed time histories using a wavelet based filterban
summation of the band-passed histories returns the original time history. This figure unfol
response time history into a very revealing display of the time-scale representation. The top le
is the mean-removed original signal, the blocks following column-wise downward are the ban
filtered signal in order of decreasing frequency, and the lower right block is the low pass chan
mean of the signal. Note the different scales on the plots for the filtered processes, indicating
contribution in that frequency band. The power spectral density of the signal in Fig. 18 is sho
Fig. 19, in which the frequency bands 1 through 7 of the filtered process are marked. The high
tive magnitudes of bands 3 and 4 correspond to the right peak in the spectrum, and are due
order wave effects. The high relative magnitude in bands 6 and 7 corresponds to structural re
due to wind and second-order wave effects. The wavelet-based filter bank has helped to ident
high frequency spikes and their time of occurrence, associated with waves slamming the st
observed in bands 1 and 2. These transient events in the response of the structure exposed to
wave fields are not clearly discernible in the time history where large excursions may be due to
occasional slamming, or large but not slamming waves. The improved efficiency over FFT and
filtering techniques, e.g., multifiltering with simple oscillators (e.g., Kameda, 1975), renders wa
filterbanks a quick and convenient time-scale decomposition method.
j i ai j,
a2i j+
j 0 1 ... i 1–, , ,=
i 1–=
32
to the
his is a
range.
ation.
ating a
ul and
the fil-
Signal Analysis with Spectral Methods and Wavelets / Time-Scale Decomposition
A wavelet power spectrum is estimated by plotting the summed coefficients with respect
scale axis only. Small changes in frequency within an octave band are not easily resolvable. T
larger problem for the high frequency range where the octave spans half the total frequency
This may be alleviated by several methods which allow intra-octave wavelet coefficient estim
One option is to apply a number of slightly dilated parent wavelets recursively to the data, cre
denser sampling grid than the octave-by-octave grid used by the original parent wavelet (Rio
Duhamel, 1992). Another method used in this study is the application of zoom techniques to
tered data.
0 500 1000
−1
0
1
2
0 500 1000−1
0
1
0 500 1000−0.5
0
0.5
0 500 1000−1
0
1
0 500 1000−0.5
0
0.5
0 500 1000−0.5
0
0.5
0 500 1000−1
0
1
0 500 1000−0.1
0
0.1
0 500 1000−1
0
1
0 500 1000−0.1
0
0.1
0 500 1000−0.5
0
0.5
0 500 1000
−1
0
1
2
Figure 18. From top left column-wise downward: measured offshorestructural response to wind and wave field and its time-scaledecomposition using wavelet transforms.
1
2
3
4
5
6
7
33
n FFT
f an off-
d speed
stimates
a under
ders its
tes are
spaced
and evo-
ay be
e shown
plitude
quen-
elocity
An example of octave band and intra-octave band wavelet spectral estimations and a
based estimation are shown in Fig. 20. The left figure is the power spectrum of the response o
shore platform to wind loads, and the right figure is the cospectrum between the measured win
input and resulting platform response. The areas under the wavelet spectral and cospectral e
represent the true variance and covariance from the time histories almost exactly, while the are
the FFT estimate does not. Further, smoothing of the FFT estimate with segment averaging ren
resolution inferior to that of the wavelet estimate at low frequencies. The FFT spectral estima
the average of 8 segments, while the wavelet estimate is based on the entire data record.
Wavelet coefficients in an octave band represent the energy at time intervals equally
over the duration of the signal, and may be used to analyze nonstationary events for transient
lutionary phenomena. Accordingly, the transfer of energy from one octave band to the next m
observed along the time scale in the scalogram. Two example applications of the scalogram ar
in Figs. 21 and 22. In Fig. 21 the analyzed signal in the top plot is a sine wave of constant am
whose frequency is steadily increased in time. The transfer of energy from lower to higher fre
cies in time is clearly demonstrated as the dark region. In Fig. 22, the signal is a hurricane v
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510
−4
10−3
10−2
10−1
100
101
Figure 19. Power spectral density of measured offshore struc-tural response seen in Fig. 18.
12345
6
7
34
record measured after the hurricane eye has passed the instrument. In the scalogram the light region
shows the band of frequency content of the record remains relatively constant, while the magnitude at
earlier time is larger than at later time, suggesting nonstationary features.
Wavelet Simulation of Nonstationary Processes
The concept of applying a modulated stationary process centered at narrow-banded frequen-
cies to model ground motion has been extensively used (e.g., Saragoni and Hart, 1974; Li and
Kareem, 1991). In this representation each component process, , is modulated by a different
modulating function
. (50)
There are different approaches to modeling and to describe .
The retention of both time and frequency information makes wavelets a useful tool for the
simulation of nonstationary signals. This can be done given either a parent nonstationary signal, or a
target spectrum and modulator function for each octave. Given a parent nonstationary signal, e.g., a
local wind velocity record, an ensemble of signals may be simulated whose average statistics closely
resemble those of the parent process. The parent signal is discrete wavelet transformed (DWT), and
the coefficients multiplied by a Gaussian white noise of unit variance . The inverse wavelet
transform (IWT) then produces a simulation statistically similar to the parent process.
Given a target spectrum and modulator functions for each octave, the simulation is done by
Figure 20. Left - Power spectrum estimates of offshore platform response using FFT, octave bandwavelet, and intra-octave band wavelet estimation techniques. Right - cospectrum esti-mates between wind velocity input and resulting platform response.
solid = FFT estimate-o- = octave band wavelet estimate....... = intra-octave band wavelet estimate
sj t( )
mj t( )
x t( ) mj t( )sj t( )j
∑=
mj sj x
w n( )
35
0 2 4 6 8 10 12 14 16 18 20
−1
−0.5
0
0.5
1
time (sec)
fre
qu
en
cy (
Hz)
0 2 4 6 8 10 12 14 16 1810
15
20
25
Figure 21. Frequency modulated sine wave and scalogram
0 10 20 30 40 50 605
10
15
20
25
30
time (sec)
fre
qu
en
cy (
Hz)
0 10 20 30 40 50
8
10
12
14
16
18
Figure 22. Measured hurricane data and scalogram
0
1
36
first finding the energy contained in each octave from the target spectrum. The wavelet coefficients
for the simulated process are multiplied with the appropriate modulator, and normalized such that the
energy equals that in the corresponding octave. These modulated and normalized coefficients are then
multiplied through by white noise and inverse wavelet transformed. The process is represented by
(Gurley and Kareem, 1994)
. (51)
When a parent signal is used to determine the modulator function, the measured wavelet coef-
ficients and target spectrum are used as
, (52)
where is a level-dependent amplitude constant and is the energy corresponding to the octave
from the target power spectrum. Figure 23 shows a measured nonstationary wind velocity record, and
a simulated process using the wavelet transform. Both statistical and visual comparisons between the
target and simulated records are good.
x n( ) IWT w n( )*mij Si
2i 2 M–+
-----------------------
=
ai j,
mi j, Ai 2i 2 M–+ ai j,
Si
----------=
Ai Si ith
Figure 23. a) Measured and b) a realization of simulatedwind velocity using wavelet transform.
0 10 20 30 40 50 605
10
15
20
25
0 10 20 30 40 50 605
10
15
20
25
a
b
37
ind on
iew of
f esti-
tion is
onlinear
Volterra
system
r simu-
lterra
plica-
tation of
accom-
s and
223),
artially
y Pro-
, Vol-
, Vol-
orks
Wiley
ll.
CONCLUDING REMARKS
Progress in quantifying and simulating the non-Gaussian and nonstationary effects of w
structures has been elusive due to the limitations of traditional analytical tools. Here, an overv
techniques is presented with examples which aid in the efficient modeling, simulation, and pd
mation of non-Gaussian processes. The estimation of Volterra kernels from system identifica
addressed, as well as other representations for nonlinear systems. The estimated pdf of n
system response is presented via several methods with examples which involve the use of
kernels in the Kac-Siegert approach, the use of joint moment information as constraints on
entropy, and the use of moment-based Hermite transformation models. Several techniques fo
lating non-Gaussian signals, including convolving Fourier amplitude pairs with higher-order Vo
kernels, nonlinear mapping, as well the concept of conditional simulation, are discussed. The im
tions of non-Gaussian winds and their load effects on fatigue damage and gust factor represen
dynamic wind loads are illustrated. The analysis and simulation of nonstationary processes is
plished by the application of localized basis functions via the wavelet transform. Application
examples are given which pertain to nonstationary effects of wind on structures.
ACKNOWLEDGEMENTS
The support for this work was provided in part by NSF Grants BCS-9096274 (BCS-8352
CMS9402196, and ONR Grant N00014-93-1-0761. The first and second authors were p
supported by a Department of Education GAANN Fellowship during this study.
REFERENCES
Ammon, D. (1990), “Approximation and Generation of Gaussian and non-Gaussian Stationarcesses,” Structural Safety, 8: 153-160.
Ang, A. H-S. and Tang, W.H. (1975), Probability Concepts in Engineering Planning & Designume I - Basic Principles. John Wiley & Sons, Inc., New York.
Ang, A. H-S. and Tang, W.H. (1984), Probability Concepts in Engineering Planning & Designume II - Decision, Risk, and Reliability. John Wiley & Sons, Inc., New York.
Antsaklis, P.J. (1993), “Control Theory Approach,” Mathematical Approaches to Neural Netw(ed. J.G. Taylor), Elsevier Science Publishers B.V., 1-23.
Bendat, J.S. (1990), Non-linear System Analysis and Identification from Random Data, Johnand Sons, Inc., New York.
Borgman, L.E. (1990), “Irregular Ocean Waves: Kinematics and Forces,” The Sea, Prentice Ha
38
glassons on9.
Wind
of a
Safe-
cribed
h ap- 28,
ST3)
lewood
Ran-
SCE,
ilizingl Con-s, Am-
l vol-hods
, Ad-
d Eng.
:226-
erna-blish-
Calderone, I., Cheung, J.C.K., and Melbourne, W.H. (1994), “The full-scale significance, oncladding panels, of data obtained from wind tunnel measurements of pressure fluctuatibuilding cladding,” Journal of Wind Engineering and Industrial Aerodynamics, 53: 247-5
Calderone and Melbourne (1993), “The Behavior of Glass Under Wind Loading,” Journal of Engineering and Industrial Aerodynamics, 48: 81-94.
Cartwright, D.E. and Longuet-Higgins, M.S. (1956), “The Statistical Distribution of the MaximaRandom Function,” Proceedings of the Royal Society of London A, Vol. 237, pp. 212-32.
Cheong, H-F. (1995), “Estimation of the minimum pressure coefficient due to gusts,” Structuralty, Elsevier, 17, 1-16.
Conner, D.A. and Hammond, J.L. (1979), “Modeling of Stochastic System Inputs Having PresDistribution and Covariance Functions,” Applied Mathematical Modeling, 3(2).
Davenport, A.G. (1964), “Note on the distribution of the largest value of a random function witplication to gust loading,” Proceedings of the Institution of Civil Engineers London, Vol.pp. 187-96.
Davenport, A.G. (1967), “Gust Loading Factors,” Journal of the Structural Division ASCE, 93(11-34.
Deutsch, R. (1962), Nonlinear Transformations of Random Processes, Prentice-Hall, EngCliffs.
Elishakoff, I., Ren, Y.J. and Shinozuka, M. (1994), “Conditional Simulation of Non-Gaussian dom Fields,” Engineering Structures, 16(7): 558-563.
Grigoriu, M. (1984), “Crossing of Non-Gaussian Translation Process,” J. Engrg. Mech., A110(4): 610-20.
Grigoriu, M. (1995), Applied non-Gaussian Processes, Prentice Hall P T R.
Gurley, K. and Kareem, A. (1994), “On the Analysis and Simulation of Random Processes UtHigher Order Spectra and Wavelet Transforms,” Proceedings of the Second Internationaference on Computational Stochastic Mechanics, Athens, Greece, Balkema Publishersterdam, Netherlands.
Gurley, K. and Kareem, A. (1995), “Simulation of a Class of Non-Normal Processes, “Speciaume of the Journal of Nonlinear Mechanics, EUROMECH-Colloquium No. 329 on Metfor Nonlinear Stochastic Structural Dynamics.
Hertz, J., Krogh, A., and Palmer, R.G. (1991), Introduction to the Theory of Neural Computationdison-Wesley Publishing Company, New York.
Holmes, J.D. (1981), “Non-Gaussian Characteristics of Wind Pressure Fluctuations,” J. of Winand Ind. Aero., 7: 103-108.
Holmes, J.D. (1985), “Wind Action on Glass and Brown’s Integral,” Engineering Structures, 7230.
Hoshiya, M. (1993), “Conditional Simulation of a Stochastic Field,” Proceedings of the 6th Inttional Conference on Structural Safety and Reliability, Innsbruck, Austria. Balkema Puers, Amsterdam, Netherlands.
39
IEEE,
Law
ering
ndom
ns.
nt
ieldspa-
U.S., LA,
essescturals.
onlin-ium on), AS-
g Plat-ME,
teady
trong
f the
g, 14:
nts on
FFT,”
Johnson, G.E. (1994), “Constructions of Particular Random Process,” Proceedings of the82(2): 270-285.
Kac, M. and Siegert, A.J.F. (1947), “On the Theory of Noise in Radio Receivers With SquareDetectors,” J. Applied Physics, Vol. 8, pp. 383-397.
Kameda, H. 1975. “Evolutionary Spectra of Seismogram by Multifilter,” Journal of the enginemechanics division, ASCE. 101(EM6): 787-801.
Kameda, H. and Morikawa, H. (1994), “A Conditioned Stochastic Processes for Conditional RaFields,” Journal of Engineering Mechanics, ASCE, 120(4): 855-875.
Kapur, J.N., (1989), Maximum Entropy Models in Science and Engineering, John Wiley and So
Kareem, A. (1978), “Wind Excited Motion of Tall Buildings,” Ph.D. Dissertation in partial fulfillmeof the degree of doctor of philosophy, Colorado State University, Fort Collins, CO.
Kareem, A. and Li, Y. (1988), “On Modeling The Nonlinear Relationship Between Random FBy Means of Higher-Order Spectra,” Probabilistic Methods in Civil Engineering (P.D. Snos, editor), ASCE, NY, pp. 384-387.
Kareem, A. (1993), “Numerical Simulation of Stochastic Wind effects,” Proceedings of the 7thNational Conference on Wind Engineering, Vol. 1, Wind Engineering Research CouncilCalifornia.
Kareem, A., Gurley, K. and Kantor, J.C. (1993), “Time-Scale Analysis of Nonstationary ProcUtilizing Wavelet Transforms,” Proceedings of the 6th International Conference on StruSafety and Reliability, Innsbruck, Austria, Balkema Publishers, Amsterdam, Netherland
Kareem, A., Hsieh, C.C., Tognarelli, M.A. (1994). “Response analysis of offshore systems to near random waves part I: wave field characteristics,” Proceedings of the special symposstochastic dynamics and reliability of nonlinear ocean systems, Ibrahim and Lin (eds.ME, Chicago, IL, Nov.
Kareem, A. and Zhao, J. (1994), “Analysis of Non-Gaussian Surge Response of Tension Leforms Under Wind Loads,” Journal of Offshore Mechanics and Arctic Engineering, ASVol. 116; 137-144.
Kawai, H. (1983), “Pressure Fluctuations on Square Prisms-Applicability of Strip and Quasi-STheories”, J. of Wind Eng. Ind. Aerodyn., 13:197-208.
Kobayashi, H., Hatanaka, A., and Ueda, T. (1994), “Active Simulation of Time Histories of SWind Gust in a Wind Tunnel,” J of Wind Engineering and Ind. Aero., 53: 315-30.
Krayer, W.R. and Marshall, R.D. (1992), “Gust Factors Applied to Hurricane Winds,” Bulletin oAmerican Meteorological Society, Vol. 73(5): 613-17.
Kung, S.Y. (1993), Digital Neural Networks, PTR Prentice Hall, Englewood Cliffs, New Jersey.
Langley, R.S. (1987), “A statistical Analysis of Non-linear Random Waves,” Ocean Engineerin389-407.
Letchford et al. (1993), “The Application of the Quasi-steady Theory to Full Scale Measuremethe Texas Tech Building,” J. of Wind Eng. Ind. Aerodyn., 48:111-132.
Li. Y. and Kareem, A., (1991), “Simulation of multivariate nonstationary random processes by
40
igital
., En-
ayer of theol. 2.
pro-
ech., Uni-
p Res.,
is ofpro-
ong-
ineer-
” J. of
Water-
f Wind
rans-
ineer-
d non-
ons.
Fluctu-: 623-
Journal of engineering mechanics, ASCE. 117: 1037-1058.
Li. Y. and Kareem, A. (1993). “Simulation of multivariate random processes: Hybrid DFT and dfiltering approach,” Journal of engineering mechanics, ASCE. 119: 1078-98.
Madsen, H.O., Krenk, S. and Lind, N.C. (1986), Methods of Structural Safety. Prentice-Hall, Incglewood Cliffs, N.J.
Maruyama, T. and Marikawa, H. (1995), “Digital Simulation of Wind Fluctuations in Boundary LUsing Experimental Data,” Wind Engineering, Retrospect and Prospect, ProceedingsNinth International Conference on Wind Engineering, New Delhi, India, January 9-13. V
Mignolet, M.P. and Spanos, P.D. (1987), “Recursive simulation of stationary multivariate randomcesses,” Journal of applied mechanics, 54: 674-87.
Murlidharan, T. and Kareem, A. (1993), “Conditional Simulation of Gaussian Velocity Field,” TReport no. CEGEOS-93-1, Department of Civil Engineering and Geological Sciencesversity of Notre Dame.
Naess, A. (1985), “Statistical Analysis of Second-order Response of Marine Structures,” J. Shi29(4): 270-284.
Nam, S.W., Powers, E.J., and Kim, S.B. (1990), “Applications of Digital Polyspectral AnalysNonlinear System Identification,”. Proc. 2nd IASTED international symposium of signal cessing and its applications. Gold Coast, Australia. pp. 133-136.
Newland, D.E. (1993), An Introduction to Random Vibrations, Spectral & Wavelet Analysis. Lman Scientific & Technical, and John Wiley & Sons, New York.
Ochi, M.K. (1986), “Non-Gaussian random processes in ocean engineering,” Probabilistic Enging Mechanics, 1: 28-39.
Peterka, J.A. and Cermak, J.E. (1975), “Wind Pressures on Buildings - Probability Densities,Structural Div., ASCE, 101 (ST6): 1255-67.
Read, W.W., and Sobey, R.J. (1987), “Phase Spectrum of Surface Gravity Waves,” Journal ofway, Port, Coastal, and Ocean Engineering, Vol. 113(5), pp. 507-522.
Reed, D.A. (1993), “Influence of Non-Gaussian Local Pressures on Cladding Glass,” Journal oEngineering and Industrial Aerodynamics, 48:51-61.
Rioul, O., and Duhamel, P. (1992), “Fast Algorithms for Discrete and Continuous Wavelet Tforms,” IEEE Transactions on Information Theory. 38: 569-86.
Saragoni, G.R., and Hart, G.C. (1974), “Simulation of Artificial Earthquakes,” Earthquake Enging Struct. Dyn., 2: 249-267.
Sarkani, S., Kihl, D.P. and Beach, J.E. (1994), “Fatigue of Welded Joints Under NarrowbanGaussian Loadings,” Probabilistic Engineering Mechanics, 9: 179-190.
Schetzen, M. (1980), The Volterra and Wiener Theories of Nonlinear Systems, John Wiley & S
Seong, S.H. and Peterka, J.A. (1993), “Computer Simulation of Non-Gaussian Wind Pressure ations,” Proceedings of the seventh U.S. National Conference on Wind Engineering, 2632.
41
J. of
oba-
Divi-
s, En-
Sys-
The-Colo-
Windonfer-
uake
ering
ds,” J.
Shinozuka, M. (1971), “Simulation of Multivariate and Multidimensional Random Processes,”Acoust. Soc. Am., 49: 357-368.
Sobczyk, K., and Trebicki, J. (1990), “Maximum Entropy Principle in Stochastic Dynamics,” Prbilistic Engineering Mechanics, 5(3): 102-110.
Soize, C. (1978), “Gust Loading Factors with Nonlinear Pressure Terms,” J. of the Structuression, ASCE, Vol. 104(6): 991-1007.
Soong, T.T., and Grigoriu, M. (1993), Random Vibration of Mechanical and Structural Systemglewood Cliffs. N.J., Prentice Hall.
Spanos, P.D. and Donley, M.G. (1991). “Equivalent Statistical Quadratization for Nonlineartems,” Journal of Engineering Mechanics, ASCE, 117(6): 1289-1309.
Tieleman, H.W., and Hajj, M.R. (1995), “Pressures on a Flat Roof-Application of Quasi-Steadyory,” Proceedings of ASCE Specialty Conference, Engineering Mechanics, Boulder, rado.
Thomas, G., Sarkar, P.P. and Mehta, K.C. (1995), “Identification of Admittance Functions for Pressures From Full-Scale Measurements,” Proceedings of the Ninth International Cence on Wind Engineering, New Delhi, India, January 9-13. pp. 1219-1230.
Vanmarcke, E.H. and Fenton, G.A. (1991), “Conditional Simulation of Local Fields of EarthqGround Motion,” Structural Safety, 10: 247-264.
Winterstein, S.R. (1988), “Nonlinear Vibration Models for Extremes and Fatigue,” J. of EngineMechanics, ASCE, 114(10): 1772-1790.
Yamazaki, F. and Shinozuka, M. (1988), “Digital Generation of Non-Gaussian Stochastic Fielof Engineering Mechanics, ASCE, 114(7): 1183-97.