directionality and uncertainty - ttu-ir.tdl.org

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
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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
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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
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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
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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
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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
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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
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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.
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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
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
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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
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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
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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
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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.
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,
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.
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;
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).
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).
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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
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:
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
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
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
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
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
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.
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
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.
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.
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
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)
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.
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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
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:
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
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.
35
UNCERTAINTY
probabilistic wind load effects (responses) of structures has been well recog

directionality and uncertainty - ttu-ir.tdl.org

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