Estimation of Probabilistic Extreme Wind load Effect with Consideration of Directionality and Uncertainty by Xinxin Zhang, B.S., M.S., A Dissertation In Wind Science and Engineering Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Approved Xinzhong Chen, Dr. Eng. Chair of Committee Kishor C. Mehta, Ph.D. Douglas A. Smith, Ph.D. Delong Zuo, Ph.D. Kathleen Gilliam, Ph.D. Mark Sheridan Dean of the Graduate School August, 2015
Estimation of Probabilistic Extreme Wind load Effect with
Consideration of
Directionality and Uncertainty
Partial Fulfillment of
the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
August, 2015
ii
ACKNOWLEDGMENTS
It has been a valuable experience for me to complete my doctoral
research.
The life during this period of time cannot be meaningful without
the care and
support from many people. I would like to present my faithful
appreciation hereby.
First and foremost, this research cannot be completed without the
guidance
from my advisor, Dr. Xinzhong Chen. His profound knowledge and
insightful
understanding of the field of study has helped the progress of the
research
remarkably. Also, his enthusiastic pursuit of academic excellence
sets up a good
example of professionalism. His financial support throughout the
study is greatly
acknowledged.
I would like to extend my appreciation to Dr. Kishor Mehta. Other
than the
timely financial support upon my arrival at the U.S. and during my
internship, I
benefit more from his broad vision in wind engineering and related
field of study.
His encouragement is important and sincerely appreciated.
I would also like to thank my committee members, Dr. Douglas Smith,
Dr.
Delong Zuo and Dr. Kathleen Gilliam for their time and suggestions
to the
research and thesis.
The financial supports from National Wind Institute of TTU, and
from
NSF Grant No. CMMI-1029922 are great acknowledged.
I am grateful to the people who showed helping hands in my
difficult times.
Special thanks are given to my dear parents who have not only
brought me
up but also shaped me with good education. Their love and care are
invaluable.
Above all, the faith, love, care, dedication and sacrifice from my
wife,
Rong Sun, cannot be listed, and my appreciation to her is beyond
words.
Texas Tech University, Xinxin Zhang, August, 2015
iii
TABLE OF CONTENTS
ACKNOWLEDGMENTS
......................................................................................................
ii ABSTRACT
.....................................................................................................................
vii LIST OF TABLES
.............................................................................................................
ix LIST OF FIGURES
............................................................................................................
xi CHAPTER 1
................................................................................................................
1
INTRODUCTION................................................................................................................
1
1.1.1. Uncertainties in the quantification of wind
effect....................................... 2
1.1.2. Consideration of directionality effect
......................................................... 3
1.1.3. Dependence of directional wind speeds
...................................................... 5
1.1.4. Challenges and motivations
........................................................................
7
1.2. Objectives and scope of the research
...............................................................
8
CHAPTER 2
..............................................................................................................
11 EXTREME WIND SPEED DATA FROM MULTIPLE SOURCES
............................................ 11
2.1. Introduction
....................................................................................................
11
2.2.1. Data
source................................................................................................
12
2.3. Analysis of inconsistency in yearly maxima combination
............................ 17
2.4. Investigation into roughness length
...............................................................
19
2.5. Directional wind speed
data...........................................................................
21
3.1. Introduction
....................................................................................................
23
3.3.1. Methods based on process upcrossing rate
............................................... 27
3.3.2. Methods based on the largest yearly wind load effect data
...................... 31
3.3.3. Storm passage method
..............................................................................
32
3.3.4. Sector-by-sector
methods..........................................................................
32
3.4. Concluding remarks
.......................................................................................
34
CHAPTER 4
..............................................................................................................
35 A UNIFIED FRAMEWORK TO CONSIDER DIRECTIONALITY AND UNCERTAINTY
............. 35
Texas Tech University, Xinxin Zhang, August, 2015
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4.2.1. Modeling of extreme wind speed at each sector
....................................... 39
4.2.2. Modeling of joint probability distribution of directional
extreme
wind speeds
...............................................................................................
41
4.3. Directional extreme wind speed model
......................................................... 49
4.3.1. Wind speed observation data
....................................................................
49
4.3.2. Extreme wind speed model from U-model
............................................... 49
4.3.3. Extreme wind speed model from Q-model
............................................... 54
4.4. Variation of directional wind load effect coefficient
..................................... 55
4.5. Comparison and validation of wind load effect estimations
......................... 58
4.6. Directionality factor
.......................................................................................
62
4.7. Consideration of uncertainties of wind load effect
coefficients .................... 63
4.8. Conclusion
.....................................................................................................
67
CHAPTER 5
..............................................................................................................
68 ON THE DEPENDENCE OF DIRECTIONAL EXTREME WIND LOAD EFFECT AND
A
SIMPLIFIED MULTIVARIATE METHOD
...........................................................................
68
5.3. Sector-by-sector method
................................................................................
74
5.5. Influence of directional wind speed masking
................................................ 79
5.6. A simplified method
......................................................................................
86
5.7. The difference between Gaussian and Gumbel copula models
..................... 90
5.8. Influence of partition of directional sectors
................................................... 92
5.9. Conclusion
...................................................................................................
101
CHAPTER 6
............................................................................................................
103
6.1. Introduction
..................................................................................................
103
6.2.1. Distribution of yearly maximum wind
speed.......................................... 105
6.2.2. Distribution of yearly maximum wind load effect
.................................. 107
6.3. Parent distribution of directional wind speeds
............................................. 110
6.3.1. Weibull distribution
................................................................................
110
Texas Tech University, Xinxin Zhang, August, 2015
v
6.4. Performance of the refined wind speed process upcrossing
rate
approach
.......................................................................................................
114
6.4.2. Estimation of wind speeds for given
MRIs............................................. 120
6.4.3. Estimation of wind load effects for given MRIs
..................................... 125
6.4.4. Influence of uncertainties in wind load effect coefficients
..................... 131
6.5. Conclusion
...................................................................................................
134
CHAPTER 7
............................................................................................................
135 CONSIDERING DIRECTIONALITY EFFECTS USING FULL-ORDER METHOD
....................... 135
7.1. Introduction
..................................................................................................
135
7.3. Directional extreme wind speed model
....................................................... 142
7.3.1. Selection of independent storms
.............................................................
142
7.3.2. Annual maximum distribution from independent storms in
each
directional sector
.....................................................................................
145
7.3.4. Estimation of directionless wind
speed................................................... 150
7.4. Estimation of wind load effect for given MRIs with
consideration of
directionality
................................................................................................
152
7.5. Conclusions and recommendation for future work
..................................... 159
7.5.1. Conclusion
..............................................................................................
159
CHAPTER 8
............................................................................................................
161 CONCLUSIONS AND FUTURE
WORKS............................................................................
161
8.1.
Conclusions..................................................................................................
161
8.1.1. Investigation of long-term wind speed record from multiple
sources .... 161
8.1.2. A multivariate framework to consider directionality and
uncertainty .... 161
8.1.3. Influence dependence of directional extreme wind load effect
.............. 162
8.1.4. A refined process upcrossing rate approach
........................................... 163
8.1.5. Considering directionality with an extension of full-order
method ........ 164
8.2. Future works
................................................................................................
165
Texas Tech University, Xinxin Zhang, August, 2015
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8.2.4. Approximation of independent number for wind load effect
................. 166
8.2.5. Assessment of overall risk of structures associated with
multiple
limit state wind-induced responses
......................................................... 166
REFERENCES
...............................................................................................................
172
APPENDIX A: YEARLY MAXIMA FROM VARIOUS DATA SOURCES
................................. 181 APPENDIX B: WIND SPEED TIME
HISTORY FOR BALTIMORE, MD ................................ 183
APPENDIX C: GEV DISTRIBUTION OF GLOBAL AND BLOCK MAXIMUM
........................ 186 APPENDIX D: CONCORDANCE AND RANK
CORRELATION ............................................. 189
APPENDIX E: INFORMATION OF STORMS IN EACH DIRECTION
...................................... 191
APPENDIX E: LIST OF SYMBOLS
..................................................................................
196 APPENDIX F: INTERDISCIPLINARY CONTRIBUTION OF THIS RESEARCH
........................ 198
Texas Tech University, Xinxin Zhang, August, 2015
vii
ABSTRACT
Estimation of wind load effects with various mean recurrence
intervals
(MRIs) requires consideration of uncertainty and directionality of
wind climate,
aerodynamics and structural dynamics. The approaches addressing
the
directionality effect are either unable to be used for parametric
study of
uncertainties of wind load effects due to lack of analytical
formulations or
inaccurate in prediction. On the other hand, the approaches to
quantify the
influence of uncertainties of wind speed and wind load effect are
not formulated
for the consideration of directionality. The objective of this
research is to establish
new and refined approaches to include the considerations of
directionality and
uncertainty in a unified framework, which also offer reasonable
predictions of
wind load effect of given MRIs.
An attempt of using long-term wind speed record is conducted for
the
purpose of reducing uncertainty of wind climate modeling.
Information of various
wind speed record resources is investigated. A reasonable choice of
wind speed
record is presented for the analysis in the following
studies.
A multivariate approach is proposed to estimate wind load effects
for
various MRIs with consideration of both directionality and
uncertainty of wind
speed and wind load effects within a unified framework. The joint
probability
distribution model of directional extreme wind speeds is
established based on
extreme wind speed data using multivariate extreme value theory
with Gaussian
Copula. The proposed approach is validated by the predictions with
those from the
existing approach. The characteristics of directionality factor for
wind load effects
are discussed. Finally, the influence of uncertainty of wind load
coefficient is
further examined.
An improved understanding of the influence of dependence
between
directional wind speeds in the estimation of probabilistic extreme
wind load effect
is provided based on the proposed multivariate approach. Several
factors that
influence the prediction with and without consideration of
dependence are
discussed. Directional wind speed masking problem is introduced and
the
Texas Tech University, Xinxin Zhang, August, 2015
viii
significance of an empirical treatment to the wind effect
estimation is discussed.
The difference between dependence structure models is discussed. A
simplified
method is proposed to reduces calculation effort. Discussion is
also made on the
partition of directional sectors.
A refined process upcrossing rate approach is introduced to improve
the
accuracy of prediction with the use of a mixed distribution model.
The
performance of the mixed distribution model is examined by
long-term
predictions in terms of wind speed and load effect. The uncertainty
effect is also
addressed. Numerical examples for buildings with various response
characteristics
demonstrate the effectiveness of the proposed framework.
An extension of a fully probabilistic method for the estimation of
extreme
wind speed of given MRIs with an additional capability of
consideration of
directionality is derived. The overall probability of exceeding a
given response
level can be regarded as a weighted sum of those in each direction.
Independent
storm maximum wind speeds are selected for the estimation of the
yearly
maximum wind speed distribution and the independent numbers. The
influence of
choice of threshold on the determination of weighting factors is
discussed. The
performance of the method is investigated by comparing the
predictions with
those determined from existing methods that based on extreme
data.
This research provides a novel solution for the structural design
concerning
the directionality and uncertainty effect. With the in-depth
investigation into the
directional dependence structure of wind speeds, this research not
only produces a
more accurate result for a risk-consistent and cost-effective
structural design but
also assists the engineers in the decision making of laboratory
tests and
interpretation of wind climate information. Better understandings
of the process
upcrossing rate approach and of the fully probabilistic methods
benefit users of
these approaches with a more accurate long-term prediction.
Moreover, the
introduction of multivariate extreme value theory enables potential
applications to
other engineering problem such as performance-based design of
structures for
multiple hazards.
ix
LIST OF TABLES
2.1 Summary of four data sources available in the United States.
............................. 14
2.2 Statistics of yearly maxima from three data sources (Duluth,
MN, 1912
to 2010)
.......................................................................................................
17
2.3 Equilibrium roughness length (m) used for calibrating wind
speeds ................... 19
2.4 Comparisons of mean yearly maximum wind speeds (mph) before
and
after calibration
...........................................................................................
20
4.1 Estimated wind speeds for each direction and regardless of
direction ................ 50
4.2 Covariance Matrices of directional wind speeds and Gaussian
variables ............ 53
4.3 Normalized wind load effect in each directions, Type I wind
speed
margins
........................................................................................................
57
4.4 Normalized wind load effects regardless of direction, example 1
Type I
wind speed margins.
...................................................................................
59
5.1 Treatment of masking problem for a piece of hourly mean wind
speed
record
..........................................................................................................
80
5.2 Comparison of type I parameters, 50- and 500-year wind speed
between
masked and full rank data
...........................................................................
83
5.3 Comparison of covariance matrix of Gaussian Variable between
masked
data and fullrank data
..................................................................................
84
5.4 Normalized directionless wind effect estimation for 16-sector
partition ............. 97
5.5 Normalized directionless wind effect estimation for 8-sector
partition,
case R0, R1 and R2
.....................................................................................
98
5.6 Normalized directionless wind effect estimation for 8-sector
partition,
case R3, R4 and R5
.....................................................................................
99
5.7 Covariance matrix of directional wind speeds for 8-sector
partition, full
rank data
......................................................................................................
99
6.2 Information of mixed distribution model and parameters of
Weibull
distribution for directional wind speeds
.................................................... 117
6.3 50- and 500-year wind speeds (mph) for directional and
non-directional
wind speeds estimated from different methods ( from 6.2)
................. 121
6.4 50- and 500-year wind speeds (mph) for directional and
non-directional
wind speeds estimated from different methods ( from )
..........................................................................................................
124
Texas Tech University, Xinxin Zhang, August, 2015
x
6.5 Normalized wind load effect calculated with various for
three
building examples (
).........................................................................
133
7.1 Comparison of statistics of selected storms with different
minimum
intervals
.....................................................................................................
144
7.2 Predicted 50- and 500- year wind speed (mph) from various
methods ............. 147
7.3 Approximation of independent numbers for each directional
sector for
distribution of directionless wind speed ( = 37 mph)
............................ 149
7.4 Approximation of independent numbers at various thresholds
for
Example 1 (b = 2)
.....................................................................................
156
7.5 Approximation of independent numbers at various thresholds
for
Example 2 (b = 2)
.....................................................................................
157
7.6 Normalized wind load effect for MRI =50 and 500 years estimated
from
different methods
......................................................................................
158
xi
LIST OF FIGURES
2.1. Yearly maximum 3-second gust from three data sources (Duluth,
MN,
1912 to 2010)
................................................................................................
16
2.2. Ratio of standardized wind speed to wind speed measured at 15
meters
for various roughness length parameters.
...................................................... 18
2.3. Yearly maximum 3-second gust from three data sources with
roughness
length correction (Duluth, MN, 1912 to 2010)
............................................. 20
3.1. Spectrum of horizontal wind speed
......................................................................
24
4.1. Relationship between correlation coefficient of Type I and its
underlying
Gaussian
........................................................................................................
45
4.2. Difference of bivariate Gaussian values with and without
consideration
of correlation
.................................................................................................
46
4.3. Multivariate extreme wind climate model with different types
of
marginal distributions (U-model)
..................................................................
51
4.4. Typical dependence structure patterns in terms of joint
probability
distributions (Type I margins)
.......................................................................
52
4.5. Multivariate extreme wind climate model with different types
marginal
distributions -model)
.................................................................................
55
4.6. Distribution of directionless wind load effect predicted from
- and -
model for both rigid and flexible structures using two approaches
.............. 60
4.7. Comparison of wind load effect distributions
...................................................... 60
4.8. Differences of predicted 50- and 500-year wind load effects
with sample
size 150
..........................................................................................................
61
4.9. Directionality factor as a function of MRI for Q- and U-models
and for
rigid and flexible structures
...........................................................................
63
4.10. Distribution of wind load effect with = 0, 0.1, 0.2, 0.3 and
0.4
(Example 1, b = 2)
.........................................................................................
65
4.11. Influence of uncertainty of wind load coefficient on
directionality
factor (b= 2)
...................................................................................................
65
4.12. Distribution of directional wind load effect with as a
function of
wind direction (b = 2)
....................................................................................
66
5.1. Influence of correlation on Gaussian variables with MRI = 10,
50 and
500 years
.......................................................................................................
78
5.2. Influence of correlation on predicted wind load effect with
MRI=50 and
500 years
.......................................................................................................
79
xii
5.3. Influence of number of correlated directions on the difference
between
wind effect calculated with and without considering correlation
(
= 0.3)
.............................................................................................................
79
5.4. 50- and 500-year wind speeds as function of direction affected
by wind
speed masking
...............................................................................................
82
5.5. Two examples of wind load coefficient as function of direction
......................... 86
5.6. Directionality changes of 500-year wind effect due to
treatment of
directional wind speed masking
....................................................................
86
5.7. Illustration of simplified procedures to calculate the
distributions of
yearly maximum wind effect (Example 1, full rank wind speed
data,
b = 2)
.............................................................................................................
90
5.8. Comparison of estimated 50-year wind effects using different
models ............... 91
5.9. Comparison of distribution of wind effects predicted from
Gaussian
copula and HK model (Example 1)
...............................................................
91
5.10. 500-year wind speeds in each direction with masked and full
rank data ........... 93
5.11. Wind load coefficient in each wind speed direction for
8-sector
partition
.........................................................................................................
95
5.12. Wind effect estimated in each direction for various rotations
(8-sector
wind speed partition)
.....................................................................................
96
6.1. Influence of threshold selection on the estimations (N
direction) ..................... 117
6.2. Influence of threshold selection on the estimations
(non-directional wind
speed)
..........................................................................................................
118
6.3. Comparison of parent distribution using Weibull and mixed
model for
each direction
..............................................................................................
120
6.4. Comparison of yearly maximum distributions using different
methods in
each direction
..............................................................................................
123
6.5. Comparison of distributions estimated using data and model
for
directionless wind speeds
............................................................................
125
6.6. Wind load coefficient as a function of direction for three
building
examples
......................................................................................................
128
6.7. Illustration of wind speed direction and building orientation
..................... 128
6.8. Yearly maximum distributions of wind load effect for three
buildings ............. 129
6.9. Comparison of normalized 50- and 500-year wind effects
calculated
from different methods for various building orientations
........................... 130
6.10. Directionality factor as a function of building orientations
and MRI for
three buildings
.............................................................................................
131
xiii
6.11. Yearly maximum distribution of wind load effect with
consideration of
both directionality and uncertainty at various levels ( ).
................ 133
7.1. Comparison of selected independent storm peaks with = 100 and
200
hours.
...........................................................................................................
144
7.2. Annual maximum wind speed distributions estimated from storm
data,
with and without thresholds ( = 100 hours).
............................................ 146
7.3. Influence of on annual maximum wind speed distributions
(Gumbel
B).
................................................................................................................
147
7.4. Comparison of annual maximum distributions of wind speed
determined
from model and non-directional data ( = 37 mph)
.................................. 150
7.5. 50-year wind speed estimated for various thresholds (Gumbel B)
.................... 151
7.6. Annual maximum distributions of directionless wind speed
determined
from model with various threshold choice
.................................................. 152
7.7. Comparison of positions of maximum hourly mean wind speed
and
maximum wind load effect within a storm (wind load
coefficient
Example 1)
..................................................................................................
153
7.8. The counting of
in the W sector for wind load effect (b = 2) ...................
155
7.9. The counting of in the W sector in terms of samples, b =
2.
..................................................................................................................
156
7.10. Comparison of annual maximum distributions of wind load
effect
estimated from full-order method and from storm passage
method
( )
...............................................................................................
159
1
1.1. Background and motivation
Wind is one of the major sources of threat to a building during its
designed
life cycle. The impact of disastrous wind is well recognized as
tornados and
hurricanes usually bring devastating casualty and economic loss to
the society,
e.g., Hurricane Katrina caused a total fatality of 1,833 and an
estimated damage of
$108 billion in the United States (Knabb et al., 2005). Also strong
winds produced
by non-disastrous weather system often play a vital role in dynamic
sensitive or
flexible structures such as high-rise buildings and long-span
bridges. Moreover,
the serviceability of a structure often concerns the safety of
accessory structures,
e.g., claddings, and the accommodation comfort, e.g., wind induced
acceleration,
which are majorly relevant to moderate wind speeds. ASCE 7-10
standard (ASCE,
2010) provides a serviceability wind speed which is related to less
strong but
more frequent wind events. The understanding of the characteristics
of wind and
wind-induced structural response is essential to a successful
design.
In a probability-based design, the capacity of a structure must
meet the
demand in term of structural response or its corresponding load,
i.e., the limit
state, which is usually represented in a probabilistic manner
(Jalayer and Cornell,
2004). Demand is determined on the basis of safety and
serviceability
consideration within a designed structural life cycle while
capacity often concerns
the economy aspect. In wind engineering, extreme wind load effect
(wind effect
for simplicity hereafter) is a general term of wind-induced
responses or their
corresponding wind loads. A performance-based design with various
safety and
economy demands often requires the computation of wind effects for
various
Mean Recurrence Intervals (MRIs) in which case the annual extreme
value
distribution of wind effect is therefore needed.
Texas Tech University, Xinxin Zhang, August, 2015
2
Building standards often provide simplified methods, usually in a
close-
form equation, to determine wind-induced response for regular
shaped buildings
with prescribed parameters according to site location, surrounding
conditions, etc.
However, it is also pointed out that for more complex buildings, a
wind tunnel
study should take place (ASCE, 2010). For a project-specific wind
tunnel study,
the wind loadings is first quantified through wind tunnel test to
reflect the
influence of terrain and wind-structural interaction and then
regarded as an input
for dynamic response analysis which reflects the influence of
structural
characteristics. Following the wind tunnel test, the extreme value
distribution of
wind effect can be estimated by further integrating wind climate
information
which can be extracted from historical meteorological record
obtained at the site
of interest.
The quantification of wind effect involves the determination
and
combination of wind climate information and aerodynamic data, both
of which
produces uncertainties. The wind climate information is represented
by wind
speeds, usually interpreted as mean hourly wind speed, contains
information of
macro-meteorological and micro-meteorological fluctuation which
indicates the
large-scale and small-scale atmospheric phenomenon. The
uncertainties of wind
speed lies variation of wind speed and wind direction and the
modeling of which
in case of scarce of data. The aerodynamic data obtained from wind
tunnel studies
describes interaction between micro-meteorological fluctuation and
structure and
it is usually represented in term of extreme wind effect load
coefficient, or simply
wind load coefficient. The uncertainties of wind load coefficient
can be attributed
to aleatory uncertainty, i.e., statistical uncertainty inherited
from the statistical
modeling of the limited data currently available and epistemic
uncertainty, i.e.,
systematic uncertainty inherited from the randomness of the
influence of terrain,
Texas Tech University, Xinxin Zhang, August, 2015
3
aerodynamic and dynamic effects. Ignoring these uncertainties may
lead to a
possible non-conservative estimation (Chen and Huang, 2010).
The first fully probabilistic method that accounts for
uncertainties of wind
speed and wind load effect conditional on wind speed was given by
Cook and
Mayne (1979, 1980) for extreme wind load effects of rigid
structures. This method
is referred to as the first-order method as it neglects the
possibility of larger wind
load effects produced by second and higher-order strongest winds in
a year
(Gumley and Wood 1982; Harris 1982). A full-order method was
proposed by
Harris (1982 and 2005) to include all orders of wind speed in
producing the same
extreme wind load effect. Chen and Huang (2010) introduced a
refined full-order
method through a much simpler derivation, which is capable of
dealing with any
type of asymptotic extreme value distribution and can be used for
both rigid and
flexible structures. It is also reported in the same literature
that the influence of
uncertainties of wind speed and wind load coefficient is mutually
dependent. The
probabilistic wind load effects have also been addressed in
literature from
different perspectives (Kareem 1987, 1988, 1990; Lutes and Sakani
2004; Bashor
and Kareem 2009; Diniz et al., 2004; Diniz and Simiu, 2005; Hanzlik
et al., 2005).
It should be emphasized that these fully probabilistic methods are
incapable of
further accounting for the effect of directionality.
1.1.2. Consideration of directionality effect
The directionality effect contains two aspects, the directionality
of wind
climate of a particular site of interest, e.g., the direction in
which the strongest
wind is likely to occur, and the directionality of wind load
coefficient, i.e., wind
load coefficient is always a function of direction due to the
sensitivity of
structures to the direction of wind load input. The importance of
considering
directionality effect in estimating probabilistic wind load effects
of structures has
well been recognized. Early attempts are based on the worst case
scenario that the
strongest wind blows from the most vulnerable direction of a
structure.
Texas Tech University, Xinxin Zhang, August, 2015
4
Calculation based on the consideration of worst case scenario is
referred to as
“upper bound method” and always leads to a conservative result.
However, as the
most unfavorable direction which produces the largest wind load and
structural
response under given wind speed does not necessarily align with the
direction of
the strongest wind, consideration of directionality effect of wind,
aerodynamics
and structural characteristics will result in a reduction of
response as compared to
the analysis regardless of direction. It is reported that the
overestimated of 50 year
loads with worst case consideration may be as high as 100% in some
cases (Irwin
et al., 2005)
In many building standards, a directionality factor is often
introduced to
consider this reduction fact. Depending on the country of the
standard used in and
the type of structure it focuses on, the directionality factor may
take different
values (Laboy-Rodriguez et al., 2014) but are all assumed to be
deterministic. For
example, in ASCE 7-10 (ASCE7-10, 2010), a directionality factor of
0.85 is
specified for claddings of buildings. This factor is only
applicable to the load
specified in code for which the calibration has been made. However,
previous
studies also found that the directionality factor should be related
MRI (Simiu and
Heckert, 1998; Rigato et al., 2001; Laboy-Rodriguez et al.,
2014).
Project-specific wind engineering studies usually take advantage of
wind
tunnel testing rather than following design codes, in which
state-of-the-art
approaches are used to directly calculate the effect of
directionality. Several
approaches have been developed in literature, some of which are
based on parent
distribution and the others on annual extreme distribution of wind
speed. The
process upcrossing rate approach initially introduced by Davenport
(1977 and
1982) remains one of the popular methods in North America. An
alternative
formulation for the crossing rate analysis was presented by Lepage
and Irwin
(1985) with a consideration of the derivative of direction variable
of wind speed.
Another approach estimates distribution of extreme wind effect
using
historical directional yearly maximum wind speed data (Simiu and
Filliben, 1981;
Texas Tech University, Xinxin Zhang, August, 2015
5
Simiu and Heckert, 1998). The storm passage method introduced in
Isyumov et al.
(2002) follows a similar scheme but directly uses the time series
of mean wind
speed and direction during storm passage instead of directional
yearly maximum
wind speed data. The sector-by-sector method is also often used due
to its
simplicity (Simiu and Filliben, 2005; Irwin et al., 2005). It
determines the extreme
response with a target MRI directly from the extreme wind speeds at
different
directions. It should be noted that none of the methods above
includes uncertainty
aspect.
1.1.3. Dependence of directional wind speeds
From the well-known Van der Hoven wind speed spectrum (Van
der
Hoven, 1957), the macro-meteorological range is centered around the
period of 4
days suggesting the averaging time of the passage of a completely
weather system
over a specific meteorology station (Harris, 1982). This
corresponds to the
statement that the annual number of independent storms is about 100
on average
(Davenport, 1968) which is later verified (Cook, 1982). Provided
that continuous
wind data measurement is available for a specific site, a storm
event is likely to
change directions during its passage. It is reported that during a
storm passage, the
wind direction may vary at least 120 degrees (e.g., Cook, 1982).
Therefore, the
extreme wind speeds in neighboring sectors often have certain level
of correlation.
An accurate estimation of wind effects should take such dependence
into
consideration. One variant of sector-by-sector method uses the
largest prediction
of all direction as the final product, which has been proved to be
non-conservative
based on the fully-correlated assumption while the other variant is
conservative
with independent assumption (Simiu and Filliben, 2005). In
Australia, the second
variant of sector-by-sector methods is used for prediction of wind
speed with
consideration of directionality. However, it is also proved that
the estimation
made by taking directional dependence into consideration should lay
in between
these the results of fully-correlated and independent assumptions
(Grigoriu, 2009).
Texas Tech University, Xinxin Zhang, August, 2015
6
The upcrossing approach treats wind speed and direction as
independent
variables while it does not offer a way to describe the dependence
between
directions. The approaches based on extreme wind speed data
(Isyumov et al.,
2002; Simiu and Filliben, 1981) implicitly included the dependence
within their
procedures, however, they are not able to quantify the influence of
such
dependence due to the lack of mathematical model.
Efforts of modeling the directional dependence have been seen in
previous
literatures. Model for angular dependence of the extreme value
distribution
parameters of different directions was proposed based on max-stable
process
models and a comparison was made (Coles and Tawn, 1991) although a
later
literature suggests that the full structure of data should be
maintained and
incorporated with a multivariate model for a more accurate
estimation (Coles and
Tawn, 1994). An early attempt to use multivariate distribution of
directional wind
speed suggests the correlations between directions are generally
weak so that an
independent assumption is appropriate but the judgment of a weak
correlation
lacks the proof from multivariate extreme value theory (Simiu et
al., 1985). Based
on the analysis of a local directional wind speed record,
directional dependence
will be more significant if directional sectors are divided by a
finer resolution
(Vega, 2008). Recent studies described the directional dependence
of wind speed
using a multivariate Gaussian translating model (Grigoriu, 2007)
based on which
an algorithm of generating large set of directional extreme wind
speed was
proposed (Grigoriu, 2009; Yeo, 2014). Multivariate extreme wind
speed models
have also been addressed in literature in terms of bivariate Gumbel
distribution
model (Simiu et al., 1984) and multivariate Gumbel distribution
models (Itoi and
Kanda, 2002).
7
1.1.4. Challenges and motivations
Despite the many methods offered to address the directionality
effect, there
is a lack of consensus between different methods. It is reported
that the estimated
structural wind-response of the former World Trade Center towers of
analysis
from two independent laboratories differs as much as 40% given that
their
aerodynamic data from wind tunnel test are similar thanks to the
improving
techniques (Irwin et al., 2005). Such difference is due to the
different methods
used in interpreting the wind tunnel data in conjunction with the
wind climate
information provided for a particular site of interest. A better
understanding of the
advantages and disadvantages of each method is needed so that the
accuracy can
be verified and further improvements can be made.
Currently, there is no reliable unified approach for dealing
with
uncertainties and directionality within a unified frame work. As
mentioned above
approach based on process upcrossing rate is unable to provide an
accurate
prediction when the process contains wind speed data that are
unrelated to the
extreme events of interest (Simiu et al., 1987) although it
contains a mathematical
formula. On the other hand, the approaches based on extreme wind
data may offer
an accurate solution but the lack of analytical expression makes it
difficult for
parameter study. Moreover among the many methods dealing with
uncertainties of
wind load coefficient none is capable of taking wind directionality
into account.
As the directionality and uncertainties both influences the
prediction but their
combined influence is yet clear, the need for a unified framework
to solve this
problem is urgent.
Although the directional dependence of wind speed is neglected by
some
of the approaches for the sake of simplicity and conservatism,
e.g., the sector-by-
sector method, its influence remains unknown. That is, how
significant is the
influence brought by directional dependence and under what
situation should it be
considered or not? The quantification of the influence of
correlation is needed to
these questions and its relationship to the parameters and
statistics of wind speed
Texas Tech University, Xinxin Zhang, August, 2015
8
is needed for a better understanding. Apparently, a mathematical
model must be
established for quantification purpose and its applicability should
be discussed.
Although the multivariate analysis offered a way to account for the
directional
dependence, the consensus of different models remains unknown.
Additionally,
the influence of directional dependence on the estimated wind
effect has not been
thoroughly studied considering various statistics of directional
wind speeds and
wind load effect coefficients. Moreover, the directional dependence
could be
affected by treatment of directional wind speed masking problem
(Vega, 2008) as
well as the partitions of directional sectors.
Last but not the least, a simplified method is desired for
engineering
practice. Ideally, such simplified method should be able to
consider directionality,
uncertainty and dependence simultaneously. Also it is better to be
able to separate
the work of meteorologist and structural engineers so that each
group of specialist
can work on their own parts which can be later combined via such
unified
framework.
1.2. Objectives and scope of the research
The main objectives of this research is to provide a better
understanding of
the existing methods for the estimation of directional wind effect
and develop
reliable and parameter-study-feasible approaches to account for
both uncertainty
and directionality of wind climate, aerodynamics and structural
characteristics in a
unified framework. In this dissertation, the consensus of wind
effect estimation
among different methods will be addressed and a multivariate
extreme wind
climate model will be proposed based on which directional wind
effect can be
estimated. Validation of the multivariate model is carried out
based on
comparative studies of different approach. Analysis of the
influence of directional
dependence associated with statistics of extreme wind speed will be
conducted
based on the verified analytical model.
The organization of this dissertation is as follows:
Texas Tech University, Xinxin Zhang, August, 2015
9
Chapter 1 introduces the background and motivation. The objective
of the
research is presented and an outline of the dissertation is
summarized.
Chapter 2 introduces and discusses the current available wind
record
sources in the United States. Standardization of wind speed
according to terrain
characteristics, high of measurement and averaging time will be
provided. The
inconsistency of the yearly maximum wind speed derived from
different sources
will be illustrated. The choice of source for wind speed record
will be provided.
Wind speed standardization will be provided and the choice of wind
speed data
source will be discussed.
Chapter 3 reviews the current existing methods that deal with
the
uncertainty and directionality effects. Comments are given to
address the merits
and drawbacks of these methods.
Chapter 4 presents a new approach of estimating wind load effects
for
various mean recurrence intervals (MRIs) with consideration of both
directionality
and uncertainty. The proposed analytical framework can be
considered as an
analytical formulation of the existing approach based on historical
directional
wind speed data, but with an additional capability of accounting
for the
uncertainty of extreme response conditional on wind speed and
direction. It can
also be regarded as an extension of the existing fully probability
methods with an
additional capability of accounting for directionality.
Applications of the proposed
approach are presented and the results are compared with those from
the existing
approach to demonstrate its accuracy. The characteristics of
directionality factor
for wind load effects are discussed. Finally, the influence of
uncertainty of
extreme response conditional on wind speed and direction is further
examined.
Chapter 5 offers a better understanding of the influence of
dependence
between directional wind speeds in estimation of probabilistic
directional wind
load effect. Several factors that influence the prediction with and
without
consideration of dependence are discussed by using Gaussian copula
model. The
influence of treatment of wind speed masking problem on the wind
effect
Texas Tech University, Xinxin Zhang, August, 2015
10
estimation is discussed. The difference of brought by dependence
structure is
discussed by a comparison between multivariate Gaussian and Gumbel
copula
models. The necessity of using multivariate approach is discussed
and a simplified
method is proposed to account for directional dependence which not
only leads to
the accurate solution but also reduces calculation effort. Also
discussion is made
on the partition of directional sectors which concerns the balance
of number of
sectors and modeling uncertainty.
Chapter 6 will introduce a refined process upcrossing rate method
with a
better modeling of strong wind speeds. The parent distribution is
modeled using a
mixed distribution in which the modest wind speeds are described by
empirical
distribution while the wind speeds in the upper tail region is
modeled by General
Pareto distribution (GPD). The choice of GPD threshold and its
influence will be
discussed. The performance of the refined method will be evaluated
through
numerical examples with respect to the estimation of directional
wind speed and
wind load effect of given MRIs. The influence of uncertainty of
wind load
coefficient on predicted wind load effect and directionality factor
will be
demonstrated.
Chapter 7 extends the full-order method to address the
directionality effect.
Derivation from the full-order method without directionality
consideration to that
with directionality consideration will be provided. The methods of
selecting of
independent storms are introduced. The effective prediction of
yearly maximum
distribution of wind speeds from storm data will be illustrated.
The determination
of independent number per year will be discussed for both wind
speed and wind
load effect prediction. Numerical examples are given to illustrate
the performance
of the methods by comparing the predictions with that from methods
based on
extreme wind speed data.
Chapter 8 summarizes the conclusions of this research and future
work is
recommended.
11
2.1. Introduction
The estimation of extreme wind load effect concerns the combination
of
aerodynamic data and wind climate information. The aerodynamic data
can be
obtained from wind tunnel testing for project-specific studies,
which describes
interaction between micro-meteorological fluctuation and structure
and is usually
represented in term of extreme wind effect load coefficient with
respect to a
particular response. Thanks to the advanced technique and
standards, such
information derived from different laboratories is usually
consistent (Irwin et al.,
2009). The wind climate information can be obtained from historical
wind speed
record at the site of interest, which contains extreme wind speed
and direction
information that is used to model its yearly extreme value
distribution. In the
assessment of probabilistic wind load effect, the variation of wind
speed has a
remarkable contribution to the quantification of uncertainties in
terms of modeling
wind climate to the data available (Chen and Huang 2010).
Current wind load standards for structural design (ASCE, 2010)
uses
approximately 15-25 years of annual maximum wind speed data,
measured at
approximately 500 stations at an averaging time of 3 seconds. This
data is used to
estimate wind speeds at return periods up to 1700 years. The short
and small
amount of data used causes large uncertainties in terms of the
extrapolation to
wind speed with large MRIs. An increase in volume of available data
may
improves the estimation of wind speed with given MRIs while the
inconsistent
historical measurements often pose difficulties for use in
climatological, statistical,
and engineering purposes (Lombardo, 2012). These inconsistencies
include
anemometer height changes, terrain conditions, averaging techniques
and
Texas Tech University, Xinxin Zhang, August, 2015
12
anemometer properties and can induce significant changes in wind
speed
magnitudes for both individual and over time events.
This chapter introduces the current available sources of wind speed
record
in the United States. Yearly maximum wind speeds recorded at
multiple
meteorological stations over time (around 100 years) from four
sources are
standardized and compared. The disagreement in wind speed
magnitudes of the
four sources is discussed. A reasonable choice of wind speed record
for the
analysis in the later chapters is made.
2.2. Obtaining and processing wind speed data
2.2.1. Data source
Currently there are four datasets that are available in the United
States for
analysis of extreme wind speeds as summarized in Table 2.1
(Lombardo, 2012;
Lombardo and Ayyub, 2014). The first data set, namely Court data
(Court, 1953)
was used primarily for estimating wind loads on temporary and
permanent
structures using data from 1912-1948.The wind speed were measured
at
anemometer heights ranging from 38 to 105 ft, which were not
corrected to a
standardized height, and recorded in monthly review issues. Minimal
information
is given on exposure of the stations, e.g., city/airport, as well
as the location
changes, i.e., when and where these anemometers were moved. A
maximum 5-min
wind speed was used for the Court data.
The second data set, titled BSS118 (Building Science Series 118),
contains
“fastest-mile” wind speed data for 129 stations in the contiguous
U.S . over the
194 ’s to 197 ’s time period (Simiu et al., 1979). Fastest-mile is
the average wind
speed obtained during the passage of one mile of wind. Depending on
the
magnitude of the wind speed, the averaging time associated with
these wind
speeds varies. The anemometer heights also varied at all stations
in BSS 118 and
are noted in Simiu et al. (1979), however wind speeds were
corrected to a
standardized 10 m height in BSS118 while assuming “open” terrain,
i.e., the
Texas Tech University, Xinxin Zhang, August, 2015
13
roughness length = 0.05 m, as all these stations were located at
airports. Cup
anemometers were used for measurement at the stations analyzed in
the study over
the entire time period. These data were used to develop previous
wind maps in
ASCE standards.
The third dataset, labeled NIST/TTU (National Institute of
Standards and
Technology/Texas Tech University) is a dataset that was used to
produce the
current wind speed maps used for wind load design (ASCE, 2010).
This data set
can be found in the website
http://www.itl.nist.gov/div898/winds/nistttu.htm. It
contains annual maximum wind speeds for 487 stations across the US
over the
196 ’s to 199 ’s time period. The anemometer height varies but the
original, raw
data was corrected to 10 m height. Although the wind speed was
noted as “peak”
gust, the averaging time varies as it was not prescribed when
post-processing the
original data. Rather it is largely dependent on the wind speed
magnitude as well
as the recording systems in place at the time of measurement, e.g.,
cup
anemometers, which are poorly documented.
The fourth dataset, ISH/ASOS (Integrated Surface
Hourly/Automated
Surface Observing System), is available at the website
ftp://ftp.ncdc.noaa.gov/pub/data/noaa/. This dataset contains wind
speed
observations from the early 197 ’s to the present day.
Approximately 1,
stations in the U.S. have sufficient wind data in order to make
long-term
projections (Lombardo, 2012). Most stations have currently set
their anemometer
heights of 10 m or 33 ft while there are a number of stations with
anemometer
height of 27 ft (NOAA manual). The terrain is assumed open with a
roughness
length 0.03m since all stations were located at airports. The
ISH/ASOS
dataset has undergone three distinct measuring periods in its
history due to
instrumentation and data averaging time changes. The first period,
before the
stations became automated (ASOS), employed no prescribed averaging
scheme
similar to the NIST/TTU data. The stations gradually changed to
ASOS in the
199 ’s including anemometer changes, although they were still cup
anemometers,
14
as well as an imposed 5-second block average window for peak wind
speed. In
the 2 ’s, all ASOS stations were then equipped with sonic
anemometers and the
peak wind speed was Calculated with a 3-second moving average
window. Due to
the different averaging time, the recorded values for peak gust may
have
remarkable difference if not properly accounted for especially for
0.03 m
(Masters et al., 2010). Although changes to the anemometer’s height
and location
were allowed, they were well-documented in history, which provided
necessary
information for further investigations.
Table 2.1 Summary of four data sources available in the United
States.
Data
Source
BSS 118 129 194 ’s-
197 ’s Varies Varies
As mentioned in the previous section, the measuring conditions,
e.g.,
anemometer height, terrain and anemometer type, as well the
averaging technique,
vary for distinct data sources and sites where the stations were
built. For use in
engineering analysis, the originally documented wind speeds are
usually
standardized with a prescribed criterion.
The consistency of yearly maximum wind speeds from different
sources
cannot be directly examined without a standardizing the respective
wind speeds as
they were measured at different conditions among which anemometer
height,
terrain condition and averaging time are the major factors that
must be made
Texas Tech University, Xinxin Zhang, August, 2015
15
consistent for all records. Terrain condition is usually
represented by the
roughness length parameter which determines vertical wind profile.
The
conversion of wind speed measured at a given anemometer height to a
nominal
height, e.g., 10 meters, can be then realized based on the vertical
wind profile.
Conversion of a wind speed concerning its averaging time can be
carried out
empirically using Durst curve (Durst, 1960). The conversion of wind
speed in
regard to its terrain and height changes can be calculated using
the logarithm law
in conjunction with a dependence of shear velocity upon prescribed
roughness
length parameters, as (Simiu and Scanlan, 1996):
( 2.1 )
Therefore
( 2.2 )
where is the standardized wind speed at height and roughness
length , is the recorded wind speed at height and roughness
length . 10 m or 33 ft, is the nominal standardized height,
is
anemometer height of measurement; 0.03 m is the standardized
roughness
length, is the roughness length for the station; and are the
shear
velocity corresponding to and respectively. Each of the data
source
provides anemometer heights. Although “Court” wind speed data did
not specify
Texas Tech University, Xinxin Zhang, August, 2015
16
the roughness length used for calculation, it can be roughly
determined from
terrain type where they were measured, e.g., the roughness length
is suggested 1
m for city and 0.1 m for open terrain in ASCE7-10 (ASCE7-10).
Averaging time
must be considered due to historical changes of reporting
intervals. 5-minute
averaging time was used in Court measurement while fastest mile
data were
reported in BSS118 and NIST/TTU sources. ASOS have a 5 second
reporting
interval. For comparison purpose, upon completing conversion with
regard to
roughness length and height, all the wind speeds are converted to
3-second gust
using Durst curve.
A long-term historical yearly maximum wind speed plot combining all
data
sets are shown in Fig. 2.1. A non-stationary trend can be observed
as the mean
wind speeds are 67.3 63.9 and 55.8 mph and standard deviations are
6.6, 8.7, and
6.9 mph for COURT, BSS118 and ASOS respectively, as summarized in
Table 2.2.
More stations are analyzed (see Appendix A) and the same
observation can be
made to each station.
Fig. 2.1. Yearly maximum 3-second gust from three data sources
(Duluth, MN,
1912 to 2010)
60
80
100
Year
17
Table 2.2 Statistics of yearly maxima from three data sources
(Duluth, MN, 1912
to 2010)
Mean (mph) 67.3 63.9 55.8 62.0
Standard deviation (mph) 6.6 8.7 6.9 8.8
Coefficient of Variation 0.098 0.137 0.123 0.143
2.3. Analysis of inconsistency in yearly maxima combination
The inconsistent standardized yearly maxima combination has already
been
noted in the previous section. Possible reasons are stated
below.
Anemometer changes from traditional mechanic type (court and
bss118) to
Sonic (ASOS). Generally the traditional measurement gives a larger
value when
averaged original data overtime due to the dynamic mechanism of
the
anemometers, i.e., step function used for calculation of wind speed
averaged over
a period of time shows that the measurement of a step up wind speed
approaches
the real value in a faster rate whereas the step down approaches
the real value in a
slower rate (Brock and Richardson, 2001).
The use of Durst curve for all site. Durst curve given in ASCE7-10
is a
representative curve to merge averaging time difference but a
site-specific curve
should be applied to achieve a more accurate conversion.
Wind speed profiles are assumed to follow a log law. The log law is
a
smooth curve to describe vertical wind profile, but wind
fluctuation indicate the
fact that the recorded wind speed at anemometer height (not equal
to 10 meters)
may have some variations from the smooth curve. When it is
converted to the
value at 10 meters, the converted wind speed may not necessarily
reflect the true
wind speed at 10 meters. Further, if the recording height was not
consistent over
the whole history (the case of this study), it will produce even
more uncertainties
to the standardized wind speed.
Texas Tech University, Xinxin Zhang, August, 2015
18
The values of roughness length were suggested to be 1 m for city
and 0.1
for open to convert Court data. However, a single roughness length
value to
represent all site condition labeled with ‘city’ is misleading.
This parameter will
vary with factors such as build up density, structure height and
distance from
build-ups to anemometer location. The same explanations can be
applied to open
terrain as well. To further illustrate this issue, the ratio of
standardized wind speed
to wind speed measured at 15 meters for various roughness length
(possible range
for city) is shown in Fig. 2.2.
Fig. 2.2. Ratio of standardized wind speed to wind speed measured
at 15 meters
for various roughness length parameters.
The range of roughness length in Fig 2.2 is a reasonable range for
‘city’
site. Consider the case that the real roughness length for a site
is 0.5m (ratio = 1.4)
where if it was treated to be 1m (ratio = 1.68), the resulting
standardized wind
speed would show 20% of amplification to what it should be.
Another phenomenon shown in Fig. 2.1 is the decreasing trend from
Court
to ASOS and even within an individual data source (especially
BSS118). It could
be explain by the change of terrain roughness as industrialization
bought
increased build-ups through the years of observation. Also, this
phenomenon is a
possible result of non-stationary climate change.
0.5 1 1.5 1.4
19
2.4. Investigation into roughness length
In the correction procedure, first, in order to simplify the task,
climate is
assumed to be stationary over years, i.e., the mean values of wind
speed from each
of the sources should show little variation. With the hypothesis
noted above, one
of the major issues in the misalignment of data is the
misinterpretation of terrain
roughness.
For court data, lack of evident description of the true terrain
roughness for
data from older sources, it is reasonable to assume a roughness
length value to
replace the value of so that the mean values of such data will line
up with the
newest data set and comply with stationarity. This assumed
roughness length is
called “equilibrium” roughness. Court wind speeds were calibrated
to have the
same mean values as BSS118 by assuming an equilibrium roughness
length. Then
the original wind speed record was once again converted.
Table 2.3 Equilibrium roughness length (m) used for calibrating
wind speeds
Station ID KDLH KWMC KISN KOKC KBTV KPIA KSHR KROW KFAT
Court 0.025 0.025 0.300 0.800 0.050 1.000 1.000 0.800 0.500
ASOS 0.073 0.090 0.048 0.019 0.116 0.137 0.034 0.046 0.028
With a fine-resolution of wind speed record from ASOS data,
current
roughness length can be calculated from turbulence intensity
(Master 2010) and
used as an “equilibrium” roughness length to modify ASOS wind
speed. Time
history of equilibrium roughness lengths were calculated for all
directions and the
mean value is regarded as the modified roughness length.
Table 2.3 lists the roughness length used for wind speed
calibration. For
Duluth, MN (KDLH), Winnemucca, NV (KWMC) and Burlington, VT
(KBTV),
court wind speeds were recorded in an open terrain. Initial
equilibrium roughness
lengths for the above sites are smaller than that of a later ASOS
record. This
allows us to make the hypothesis that build-ups have been developed
through the
Texas Tech University, Xinxin Zhang, August, 2015
20
years. Calibrated time history of wind speeds for Duluth, MN is
shown in Fig. 2.3.
Although the mean value of wind speeds from three sources can be
lined up, the
calibration has not made the variation of wind speed consistent
over time.
Therefore, a more detailed and comprehensive study to the data is
needed for
further improvements.
Fig. 2.3. Yearly maximum 3-second gust from three data sources with
roughness
length correction (Duluth, MN, 1912 to 2010)
Table 2.4 Comparisons of mean yearly maximum wind speeds (mph)
before and
after calibration
Source Court BSS ASOS Court BSS ASOS Court BSS ASOS
Standard 72.0 62.2 57.9 69.6 60.9 57.5 88.0 60.9 63.3
Equilibrium 62.1 62.2 62.5 60.4 60.9 63.2 66.1 60.9 64.7
Station KOKC KBTV KPIA
Source Court BSS ASOS court BSS ASOS court BSS ASOS
Standard 84.0 66.1 66.1 61.2 57.3 53.1 68.0 63.6 59.9
Equilibrium 78.7 66.1 61.2 56.4 57.3 62.5 68.0 63.6 71.8
Station KSHR KROW KFAT
Source Court BSS ASOS Court BSS ASOS Court BSS ASOS
Standard 78.9 72.8 64.2 74.5 69.5 61.6 54.5 44.6 45.1
Equilibrium 78.9 72.8 62.9 70.8 69.5 63.1 47.5 44.6 43.3
1900 1920 1940 1960 1980 2000 2020 40
50
60
70
80
90
Year
21
2.5. Directional wind speed data
The data chosen for this dissertation is the ASOS wind speed
record
observed at Baltimore, MD, USA (Station ID: KBWI), dated from
January 1 st ,
2000 to August 31 st , 2012. Two months of wind speed data were
discarded to
exclude the influence of hurricanes, i.e., Hurricane Isabel in
September, 2003 and
Hurricane Irene in August, 2011, according to historical record.
The measurement
height is at 33 ft throughout the time of observation. The raw data
contains wind
speed in unit of knot and a wind direction with a resolution of 1 .
The hourly
mean wind speed and direction is then calculated using a vector
average algorithm.
After post-processing, the mean wind speed is in unit of miles per
hour (mph) and
wind direction has a resolution of 1 . The hours containing bad
records due to
failure of instruments or maintenance were purged. Therefore, there
are altogether
101,551 pairs of hourly mean wind speed and direction
(approximately 11.6 years)
that survive the quality control for further studies. Time index of
hours is also
kept for reference. Also noted is that the wind directions are
recorded in
meteorological coordinate with 0 or 360 representing true north and
increase
clockwise.
The wind speeds is then categorized into directional sectors that
are
evenly divided and the -th sector is represented by center
direction with an
angular width from – 180 / to + 180 / . For example, when 8 sectors
are
divided, the North directional sector includes all the wind speeds
with wind
direction ranged from 0 to 22.5 and from 337.5 to 360 , and all
wind speeds
within this range are assumed to blow from North. The partition of
wind speed
sector could be with finer resolutions but at the cost of having
fewer observations
per sector. The influence of the number of partitions will be
discussed in the later
chapters.
22
2.6. Conclusion
The information of current available sources of wind speed record
in the
United States is provided and compared in terms of available years,
terrain
characteristics, heights of measurement, averaging time and
anemometer type.
Standardization of wind speeds over time is made according to the
information
provided. A non-stationary trend is observed when standardized wind
speeds from
various sources are combined together. A correction of roughness
length
information is carried out for an improvement but without future
comprehensive
investigation of the data, the use of long-term wind speed
information from
multiple sources may not be a viable solution to reduce the
uncertainty brought by
short record. A wind speed record based on most recent stage of
ASOS
observation is chosen and processed for the analysis in the
following chapters.
Texas Tech University, Xinxin Zhang, August, 2015
23
3.1. Introduction
This chapter introduces and reviews the existing methods that
quantify the
uncertainty and directionality effects with a better understanding.
The first-order
and full-order method account for uncertainty through a fully
probabilistic
perspective. The methods accounting for directionality includes the
process
upcrossing rate approach, the methods based on yearly and storm
maximum wind
load effect and the sector-sector-methods. The advantages and
drawbacks of each
method are discussed.
3.2. First-order and full-order methods
The extreme wind load effect over a period of time , can be
calculated
from extreme wind load coefficient and the mean wind speed within
the same
time period as:
( 3.1 )
where is the air density which can be assumed to be a constant at
strong wind
condition. According to the well known spectral gap in the
wide-range frequency
spectrum of the nature wind as shown in Fig. 3.1 (Van der Hoven,
1957), for
structural design, is practically designated as one hour or ten
minutes with the
purpose of eliminating the micro-meteorological fluctuation of wind
speed while
the macro-meteorological fluctuation, i.e., the variation of mean
wind speed, is
retained (Cook and Mayne, 1979; Harris, 1982). In this study, = 1
hour is used.
Texas Tech University, Xinxin Zhang, August, 2015
24
Fig. 3.1. Spectrum of horizontal wind speed
and , obtained from wind speed record and wind tunnel testing under
a
given mean wind speed respectively, are generally random
quantities. Hence is
inherently a random variable whose parent distribution can be
quantified by
combining the probabilities of wind speed and wind load coefficient
as (Harris
2005, Chen and Huang 2010):
∞
Where is the parent cumulative distribution function (CDF) of
,
is the probability density function of and is the CDF of . Based
on
the asymptotic extreme value theory, the annual maximum
distribution of can
be determined as
25
where is the number of independent wind load effect per year. Eqs.
(3.2) and
(3.3) are fundamental while not of practical use for the
calculation of the
probabilistic wind load effect due to the difficulties in the
determination
and . requires the continuous wind speed record which, in some
cases, is
unknown. Even if it is known, the estimation of may not well
represent
statistics of strong winds important for structural design as it
may be affected to a
large extent with meteorological phenomena, e.g., morning breeze
(Simiu and
Scanlan, 1996). Hourly mean wind speeds within a storm system are
correlated so
that the value of is far smaller than 8760 hours per year and is
generally
unknown with sufficient precision (e.g., van der Hoven, 1957; Cook,
1982;
Gumley and Wood, 1982; Simiu and Heckert, 1996). In practice, a
reliable
estimation of probabilistic annual maximum wind effect has to be
performed
based on the distribution of annual maximum wind speed.
The first-order method calculates the annual maximum wind load
effect as
(Cook and Mayne, 1980):
( 3.4 )
where the is the PDF of annual maximum distribution of wind speed,
which
is determined directly from the sample of the largest wind speeds
within a year.
generally follows Gumbel distribution as suggested in the
original
literature (Cook and Mayne, 1980). Eq. (3.4) assumes that the
largest wind load
effect always comes from the largest wind speed in the a year while
neglects the
fact that a higher order wind speed in the same year may result in
a larger
response due to the variation of wind load coefficient.
The full-order method was initially proposed by Harris (1982) with
an
requirement of Gumbel distribution for , and , and was later
improved to be
applicable for all types of extreme value distributions by Harris
(2005). It
Texas Tech University, Xinxin Zhang, August, 2015
26
considers the probability the largest response produced by all
orders of wind
speeds as:
( 3.5 )
where is annual PDF the -th largest wind speed of independent
values of in a year.
The derivation of Eq. (3.5) is very complicated. However, an
alternative
formulation is provided by Chen and Huang (2010) using a simpler
derivation,
which will be presented here. The asymptotic extreme value
distributions have the
following properties in terms of their relationship with parent
distributions (e.g.,
Kotz and Nadarajah, 2000):
( 3.7 )
Multiplying the independent number on both sides of Eq. (3.2)
and
invoking Eq. (3.6) and (3.7) the following equation can be derived
(Chen and
Huang, 2010):
27
The above equations are based on the modeling of wind speed ,
termed as
-model while if the distribution of square of or is used to
substitute
the distributions of in the above equations, it is termed as
-model.
A comprehensive study using Eq. (3.8) to quantify the influence
of
uncertainties of wind speed and wind load coefficient was carried
out (Chen and
Huang, 2010), conclusion of which includes but is not limited
to:
1) Predictions of wind load effect with large MRIs are sensitive to
the
upper tail behavior of annual maximum distribution of wind load
effect .
2) The first-order method does not differ much than the full order
method
in the prediction of wind load effect for large MRIs.
3) The wind load effect for a given MRI increases with the increase
in
variations of wind speed and extreme wind load coefficient while
the significance
of the variation of one of these two on the wind load effect
depends on the
variation of another.
4) The wind load effect for a given MRI is more sensitive to the
variation
of wind speed than to that of wind load coefficient.
Although the first- and full-order methods offer rational
quantifications of
uncertainty effect associated with structural design, they are not
capable of
accounting for the effect of directionality.
3.3. Methods accounting for directionality
3.3.1. Methods based on process upcrossing rate
The upcrossing rate of the random process at the level can be
calculated as follows based on Rice formula (Rice, 1944) as:
( 3.9 )
28
.
In the case of Gaussian process, the process and its derivative
process are
independent and the derivative process follows a Gaussian
distribution. Eq. (3.9)
becomes:
( 3.10 )
where
is the mean upcrossing crossing (cycling) rate at the mean
level; and are the standard deviations of and ; is the
probability density function (PDF) of .
However, the wind speed process generally follows a Weibull
distribution.
Eq. (2.10) is not valid for a general non-Gaussian process.
Nevertheless, Eq. (3.10)
was introduced by Davenport (1977) to approximately estimate the
crossing rate
of wind speed process. The effectiveness of this approximation was
illustrated by
Grigoriu (1984) based on the translation process theory. It was
suggested to
replace the constant by (Gomes and Vickery, 1977; ASCE,
1999).
The mean cycling rate which can be determined through analysis in
frequency
domain and generally lies in the range of 500 - 1000 cycles/year or
1-2 cycles per
day for wind speed (Davenport, 1977).
To account for directionality in the calculating the upcrossing
rate of a
given wind load effect (response) level, , it is usually done by
first transfer the
response into a two-dimensional wind speed boundary calculated
as:
( 3.11 )
where is the hourly mean wind speed required to produce wind load
effect
in direction ; and is normalized extreme load effect
(response)
Texas Tech University, Xinxin Zhang, August, 2015
29
coefficient within one hour, or simply referred to as extreme load
coefficient
which can be obtained from a wind tunnel test. In the case of rigid
structures, the
load effect coefficient can be independent of mean wind speed. On
the
other hand, it is a function of mean wind speed for flexible
structures due to
dynamic amplification effect.
The upcrossing rate of is identical to the upcrossing rate of wind
speed
process at two-dimensional boundary , which can be resort to the
upcrossing
rate calculation of a vector process where wind velocity is
regarded as a two
dimensional variable (Davenport, 1977):
where is the JPDF of wind speed and direction;
is the PDF of wind speed conditional on wind direction ; and
is
PDF of wind direction, which can be estimated as the ratio of the
number of wind
speeds in direction to the total number regardless of
direction.
An alternative formulation was later developed with a slight
difference as
(Lepage and Irwin, 1985):
d ( 3.13 )
( , ).
30
Upon the determination of for various values of , the
cumulative
distribution function (CDF) of wind effect can be determined based
on Poisson
assumption as:
( 3.14 )
where is time and equals to one year, i.e., when hourly mean
wind speed process is used for crossing analysis; is the MRI in
years; and is
the wind speed with a MRI of years.
The parent distribution of wind speed in each direction is usually
modeled
by Weibull distribution based on full record of observations.
Although the
Weibull distribution generally results in a good approximation to
the modest wind
speed around mean value, it does not describe the strong wind
speeds in the upper
tail region very well, which in turn can be inaccurate in the
estimation of wind
load effect of large MRIs. The limitation of using Weibull
distribution for
calculating the upcrossing rate of wind speed process and for
determination of
distributions of annual maximum wind speed and wind load effect has
been
demonstrated in literature (Wen, 1983 and 1984; Simiu and Scanlan,
1996; Irwin
et al., 2005). A double Weibull distribution was proposed for a
better description
of the upper tail region (Xu et al., 2008; Isyumov et al., 2014).
This continuous
distribution is a weighted combination of two Weibull distributions
fitted using
log-scale-error-minimization, i.e., more weight towards the tail,
and linear scale-
error-minimization, i.e., more weight towards the mean,
respectively.
Nevertheless, the theoretically rigorous upcrossing rate approach
should be
used with cautious unless a more sensible distribution that better
reflects the
characteristics of wind speed in the upper tail region is
used.
Texas Tech University, Xinxin Zhang, August, 2015
31
3.3.2. Methods based on the largest yearly wind load effect
data
One of the most popular approaches based on extreme wind speed
data
(Simiu and Scanlan, 1996) will be briefly introduced in this
section. Denote as
the yearly maximum hourly mean wind speed in the -th direction for
the -th year
where 1, 2, …, and 1, 2, …, . The corresponding extreme wind
load
effect can be calculated as:
( 3.15 )
where is wind load coefficient in the -th direction. The maximum
wind load
effect of the -th year = max { } is then determined. The
equivalent wind speed is defined as:
( 3.16 )
where . The sample of is then fitted by Type I distribution
denoted as . The yearly maximum distribution of wind load effect
is
related to it as where and the wind load effect
for MRI = R year is then determined as
This approach considers the directionality effect using extreme
wind load
effect directly and the prediction can be reasonable. However, this
approach treat
the wind load coefficient as a deterministic value in each
direction and due to the
lack of analytical formulation to integrate distributions of wind
speed and wind
load coefficient, it is incapable of quantifying the uncertainty
effect directly. Also,
the dependence of extreme directional winds is only implicitly
accounted for as
the maximum wind effects may come from different directions rather
than a
Texas Tech University, Xinxin Zhang, August, 2015
32
mathematical model. Therefore, it is difficult to conduct a
comprehensive study
the influence brought by directional dependence.
3.3.3. Storm passage method
The storm passage method has become a standard practice in
Boundary
layer Wind Tunnel Laboratory for estimation of wind load effect,
especially for
those in hurricane-prone regions (Isyumov et al., 2014; Simiu and
Scanlan, 1996).
This method is somewhat similar to the method mentioned in the
above section
except for that the largest yearly wind load effect is replaced by
the largest wind
load effect within a storm.
For each storm, the equivalent wind speed, , can be obtained using
Eq.
(3.15) and (3.16). The parent CDF of storm maximum response, , can
be
determined and denoted as . Then the annual maximum
distribution
of wind load effect can be determined as:
( 3.17)
Compared to the method in 3.3.4, the storm passage method uses
more
extreme data and therefore reduces the modeling uncertainty.
However, like the
method in 3.3.4, it has not analytical formulation for
comprehensive parametric
study.
3.3.4. Sector-by-sector methods
A simple procedure, termed as the sector-by-sector method, can be
used for
any type of structures (Simiu and Filliben, 2005, Irwin et al.,
2005). The
probability of a particular response not exceeding a level can be
found as:
Texas Tech University, Xinxin Zhang, August, 2015
33
( 3.18)
Where is the wind speed in the -th direction that produces . In the
case that
directional extreme wind speeds in each direction are fully
correlated, Eq. (3.18)
can be reduced to:
where denotes the direction with .
Eq. (3.19) corresponding to the first variant of the
sector-by-sector method. It
implies that under the fully correlated situation, the wind load
effect of a given
MRI equal to the wind load effect of the same MRI in the -th
direction. However,
it can be proved that when the fully correlation assumption is not
valid, Eq. (3.19)
can underestimate the risk. Consider the case of MRI = 50 year,
using Eq. (3.19),
0.98. If the wind speeds are mutually
independent, one can found from Eq. (3.18):
( 3.20)
In such a case, , i.e., corresponds to an MRI≤ 5 years which
indicate an underestimation of the risk. In fact, Eq. (3.20) is
proved to always
result in a conservative prediction, and it can be regarded as the
second variant of
the sector-by-sector method.
The sector-by-sector method recognizes the dependence of
directional
wind speeds either as fully correlated or mutually independent, the
latter of which
is conservative. However, these two variants only construct bounds
for the
Texas Tech University, Xinxin Zhang, August, 2015
34
predictions with a general dependence as discussed in Chapter 5,
and therefore
may produces less accurate estimation.
3.4. Concluding remarks
reviewed. The first-order and full-order methods quantify the
uncertainty of wind
speed and wind load coefficient and agree with each other when MRI
is large but
they are not capable of accounting for directionality effect. The
process
upcrossing rate approach accounts for directionality effect while
it may fall short
in an accurate estimation unless a reasonable distribution for the
strong winds is
used. The methods based on extreme wind load effect data, including
largest
yearly and storm maximum wind load effect, accounts for
directionality effect
more accurately while they are not convenient to carry out
parameter studies as it
is not formulated in an analytical form. The sector-by-sector
method is simple to
use but it may be less accurate.
Texas Tech University, Xinxin Zhang, August, 2015
35
UNCERTAINTY
probabilistic wind load effects (responses) of structures has been
well recog