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Ming-Yi LiuMing-Yi Liu
Yu-Jie LiYu-Jie Li
Reliability Analysis of High-Rise Buildings Reliability Analysis of High-Rise Buildings
under Wind Loadsunder Wind Loads
Department of Civil Engineering, Chung Yuan Christian University, TaiwanDepartment of Civil Engineering, Chung Yuan Christian University, Taiwan
2The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Hurricane Katrina, August 2005Hurricane Katrina, August 2005
Hyatt Regency New Orleans, USAHyatt Regency New Orleans, USA
Structural Safety Affected by WindStructural Safety Affected by Wind
3The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Taipei 101, Taiwan, Completed in 2004, 508 m HighTaipei 101, Taiwan, Completed in 2004, 508 m High
Tuned Mass DamperTuned Mass Damper
Occupant Comfort Affected by WindOccupant Comfort Affected by Wind
4The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
ObjectivesObjectives The objective of this paper is to conduct the reliability analysis of The objective of this paper is to conduct the reliability analysis of
high-rise buildings under wind loads. Numerical examples are high-rise buildings under wind loads. Numerical examples are provided to capture the dynamic effects of structures with provided to capture the dynamic effects of structures with eccentricity between the elastic and mass centers. The framework of eccentricity between the elastic and mass centers. The framework of this research consists of two stagesthis research consists of two stages
The first stage includes two parts: the deterministic analysis of The first stage includes two parts: the deterministic analysis of wind-induced acceleration for a variety of attack angles, i.e., the wind-induced acceleration for a variety of attack angles, i.e., the demand, and the determination of allowable acceleration based on demand, and the determination of allowable acceleration based on the occupant comfort criteria for wind-excited buildings, i.e., the the occupant comfort criteria for wind-excited buildings, i.e., the capacitycapacity
According to the results obtained in the first stage, the reliability According to the results obtained in the first stage, the reliability analysis is conducted in the second stage, which can predict the analysis is conducted in the second stage, which can predict the probability of dissatisfaction with occupant comfort criteria for a probability of dissatisfaction with occupant comfort criteria for a variety of probability distributions of the structural eccentricityvariety of probability distributions of the structural eccentricity
5The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
FrameworkFramework
Cross-spectral density functionof wind velocity
Cross-spectral density functionof wind load
Root-mean-square accelerationat corner
Wind velocity profile
Wind load model
Cross-spectral density functionof acceleration
Frequency response functionof acceleration
Mass, stiffness and damping matrices
High-rise building model
Peak factor
Root-mean-square accelerationat mass center
Peak accelerationat corner
Demand(frequency domain analysis)
Allowable peak acceleration
Capacity(occupant comfort criteria)
The first stage
Probability of dissatisfactionwith occupant comfort criteria
Rackwitz-Fiessler method Finite difference method
Design point
Reliability analysis(synthetic method)
The second stage
Reliability index
High-rise High-rise building modelbuilding model Wind load modelWind load model
DemandDemand
CapacityCapacity
Reliability Reliability analysisanalysis
6The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
High-Rise Building ModelHigh-Rise Building Model
Cross-spectral density functionof wind velocity
Cross-spectral density functionof wind load
Root-mean-square accelerationat corner
Wind velocity profile
Wind load model
Cross-spectral density functionof acceleration
Frequency response functionof acceleration
Mass, stiffness and damping matrices
High-rise building model
Peak factor
Root-mean-square accelerationat mass center
Peak accelerationat corner
Demand(frequency domain analysis)
Allowable peak acceleration
Capacity(occupant comfort criteria)
The first stage
Probability of dissatisfactionwith occupant comfort criteria
Rackwitz-Fiessler method Finite difference method
Design point
Reliability analysis(synthetic method)
The second stage
Reliability index
High-rise High-rise building modelbuilding model
7The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
NN-Story Torsionally Coupled System-Story Torsionally Coupled SystemElasticCenterAxis
MassCenterAxis
ith Floor
2nd Floor
Nth Floor
(N-1)th Floor
1st Floor
Di
BiHi
Zi
Aerodynamic CenterAxis
θixi
yi
θ
y
x
z
Mass centerMass centerElastic centerElastic center
Aerodynamic centerAerodynamic center
Three-dimensional configurationThree-dimensional configuration
MCi
ECi
ACi
x
y
Di
Bi
Ayi
Axi
Eyi
Exi
θ
Top view of the ith floorTop view of the ith floor
Mass centerMass centerElastic centerElastic center
Aerodynamic centerAerodynamic center
8The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Wind Load ModelWind Load Model
Cross-spectral density functionof wind velocity
Cross-spectral density functionof wind load
Root-mean-square accelerationat corner
Wind velocity profile
Wind load model
Cross-spectral density functionof acceleration
Frequency response functionof acceleration
Mass, stiffness and damping matrices
High-rise building model
Peak factor
Root-mean-square accelerationat mass center
Peak accelerationat corner
Demand(frequency domain analysis)
Allowable peak acceleration
Capacity(occupant comfort criteria)
The first stage
Probability of dissatisfactionwith occupant comfort criteria
Rackwitz-Fiessler method Finite difference method
Design point
Reliability analysis(synthetic method)
The second stage
Reliability index
Wind load modelWind load model
9The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Wind Load ComponentsWind Load Components
MCi
ECi
ACi
x
y
Di
Bi θ
Drag
Lift
Torque
Wind Direction
DragDrag
LiftLift
TorqueTorque
Attack angleAttack angle
Wind directionWind direction
Top view of the ith floorTop view of the ith floor
10The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
DemandDemand
Cross-spectral density functionof wind velocity
Cross-spectral density functionof wind load
Root-mean-square accelerationat corner
Wind velocity profile
Wind load model
Cross-spectral density functionof acceleration
Frequency response functionof acceleration
Mass, stiffness and damping matrices
High-rise building model
Peak factor
Root-mean-square accelerationat mass center
Peak accelerationat corner
Demand(frequency domain analysis)
Allowable peak acceleration
Capacity(occupant comfort criteria)
The first stage
Probability of dissatisfactionwith occupant comfort criteria
Rackwitz-Fiessler method Finite difference method
Design point
Reliability analysis(synthetic method)
The second stage
Reliability index
DemandDemand
11The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Frequency Domain AnalysisFrequency Domain Analysis
Root-mean-square accelerationat corner
Cross-spectral density functionof acceleration
Peak factor
Root-mean-square accelerationat mass center
Peak accelerationat corner
Demand(frequency domain analysis)
12The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
CapacityCapacity
Cross-spectral density functionof wind velocity
Cross-spectral density functionof wind load
Root-mean-square accelerationat corner
Wind velocity profile
Wind load model
Cross-spectral density functionof acceleration
Frequency response functionof acceleration
Mass, stiffness and damping matrices
High-rise building model
Peak factor
Root-mean-square accelerationat mass center
Peak accelerationat corner
Demand(frequency domain analysis)
Allowable peak acceleration
Capacity(occupant comfort criteria)
The first stage
Probability of dissatisfactionwith occupant comfort criteria
Rackwitz-Fiessler method Finite difference method
Design point
Reliability analysis(synthetic method)
The second stage
Reliability index
CapacityCapacity
13The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Occupant Comfort CriteriaOccupant Comfort Criteria
Melbourne and Palmer Melbourne and Palmer (1992)(1992)
DDuration of uration of wind velocitywind velocity
Frequency of Frequency of structural oscillationstructural oscillation
RReturn period of eturn period of wind velocitywind velocity
Occupant Occupant comfort criteriacomfort criteria
14The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Reliability AnalysisReliability Analysis
Cross-spectral density functionof wind velocity
Cross-spectral density functionof wind load
Root-mean-square accelerationat corner
Wind velocity profile
Wind load model
Cross-spectral density functionof acceleration
Frequency response functionof acceleration
Mass, stiffness and damping matrices
High-rise building model
Peak factor
Root-mean-square accelerationat mass center
Peak accelerationat corner
Demand(frequency domain analysis)
Allowable peak acceleration
Capacity(occupant comfort criteria)
The first stage
Probability of dissatisfactionwith occupant comfort criteria
Rackwitz-Fiessler method Finite difference method
Design point
Reliability analysis(synthetic method)
The second stage
Reliability index
Reliability Reliability analysisanalysis
15The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Synthetic MethodSynthetic Method
Probability of dissatisfactionwith occupant comfort criteria
Rackwitz-Fiessler method Finite difference method
Design point
Reliability analysis(synthetic method)
Reliability index
16The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Limit State Function and Basic VariablesLimit State Function and Basic Variables
0Z
0Z
0ZLimit state
Unsafe region
Safe region
Limit state function
1X
2X nXXXgZ ,,, 21 Basic variable
Basic variable
0Z
0Z
0Z
1X
2X nXXXgZ ,,, 21
*,*,*, 21 nxxx
Design point
Reliability index
Basic variable
Basic variable
Unsafe regionLimit state
Safe region
Limit state function
Original coordinate systemOriginal coordinate system Transformed coordinate systemTransformed coordinate system
Limit state functionLimit state function
Basic variableBasic variable
Limit state functionLimit state function
Basic variableBasic variable
Basic variableBasic variable Basic variableBasic variable
Design pointDesign point
Reliability indexReliability index
17The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Numerical ExamplesNumerical Examples Two numerical examples, i.e., the torsionally uncoupled and coupled Two numerical examples, i.e., the torsionally uncoupled and coupled
systems (40-story buildings), are provided to conduct the reliability systems (40-story buildings), are provided to conduct the reliability analysis of high-rise buildings under wind loads for a variety of analysis of high-rise buildings under wind loads for a variety of attack anglesattack angles
Four types of parameters: the high-rise building model, wind load Four types of parameters: the high-rise building model, wind load model, occupant comfort criteria and reliability analysis, are model, occupant comfort criteria and reliability analysis, are considered in this studyconsidered in this study
All parameters of the two numerical examples are the same except All parameters of the two numerical examples are the same except the eccentricity between the elastic and mass centersthe eccentricity between the elastic and mass centers
Three types of probability distributions: the normal, lognormal and Three types of probability distributions: the normal, lognormal and type I extreme value distributions, are used to model the type I extreme value distributions, are used to model the uncertainties of the eccentricity between the elastic and mass centersuncertainties of the eccentricity between the elastic and mass centers
18The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Cross-Spectral Density Function of Wind LoadCross-Spectral Density Function of Wind Load
10-4
10-2
100
102
104
100
102
104
106
108
1010
1012
ω (rad/sec)
S F
xixi(ω
) (
N2 -s
ec/r
ad)
40th Floor30th Floor20th Floor10th Floor
10-4
10-2
100
102
104
100
102
104
106
108
1010
1012
ω (rad/sec)
S F
yiy
i(ω)
(N
2 -sec
/rad
)
40th Floor30th Floor20th Floor10th Floor
10-4
10-2
100
102
104
100
102
104
106
108
1010
1012
ω (rad/sec)
S F i
i (
ω)
(N
2 -sec
/rad
)
40th Floor30th Floor20th Floor10th Floor
10-4
10-2
100
102
104
100
102
104
106
108
1010
1012
ω (rad/sec)
S F
xiy
i(ω)
(N
2 -sec
/rad
)
40th Floor30th Floor20th Floor10th Floor
θθθθ
xxxx yyyy
xxyy
Attack Angle = 45˚Attack Angle = 45˚
19The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Cross-Spectral Density Function of Acceleration Cross-Spectral Density Function of Acceleration
(1)(1)
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. xi
..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. yi..(
ω)
(m
2 /sec
3 -rad
)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. xi
..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. yi..(
ω)
(m
2 /sec
3 -rad
)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S R i
.. i..(
ω)
(ra
d/se
c3 )
40th Floor30th Floor20th Floor10th Floor The first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. y
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S R i
.. i..(
ω)
(ra
d/se
c3 )
40th Floor30th Floor20th Floor10th Floor The first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. y
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi..
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. i..(
ω)
(m
2 /sec
3 )
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi..
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. i..(
ω)
(m
2 /sec
3 )
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
xxxx yyyy θθθθ
xxyy xxθθ yyθθ
11 3344 55 6677 8899 101011111212 22
11 44 77 99 1212 11 44 77 99 121255 88 111122
55 88 11112233 66 1010 33 66 1010
Torsionally Uncoupled System, Torsionally Uncoupled System, Attack Angle = 45˚Attack Angle = 45˚
20The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Torsionally Coupled System, Torsionally Coupled System, Attack Angle = 45˚Attack Angle = 45˚
Cross-Spectral Density Function of Acceleration Cross-Spectral Density Function of Acceleration
(2)(2)
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. xi
..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. yi..(
ω)
(m
2 /sec
3 -rad
)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. xi
..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. yi..(
ω)
(m
2 /sec
3 -rad
)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S R i
.. i..(
ω)
(ra
d/se
c3 )
40th Floor30th Floor20th Floor10th Floor The first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. y
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S R i
.. i..(
ω)
(ra
d/se
c3 )
40th Floor30th Floor20th Floor10th Floor The first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi.. y
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi..
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. i..(
ω)
(m
2 /sec
3 )
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Rxi..
i..(ω
) (
m2 /s
ec3 -r
ad)
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
0 5 10 15 20 25 3010
-20
10-15
10-10
10-5
ω (rad/sec)
S Ry
i.. i..(
ω)
(m
2 /sec
3 )
40th Floor30th Floor20th Floor10th FloorThe first ten naturalfrequency of structure
xxxx yyyy θθθθ
xxyy xxθθ yyθθ
11 44 77 99 1212 55 88 111122 33 66 101033 66 1010 33 66 1010 11 44 77 99 1212
55 88 111122
11 44 77 99 1212 11 44 77 99 121255 88 111122
33 66 101033 66 1010 33 66 1010
55 88 111122
55 88 111122
11 44 77 99 1212
21The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
0.02
0.04
0.06
0.08
0.1
30
210
60
240
90
270
120
300
150
330
180 0
Unit: m/sec2
CapacityDemand
0.0627
0.02
0.04
0.06
0.08
0.1
30
210
60
240
90
270
120
300
150
330
180 0
Unit: m/sec2
CapacityDemand
0.0625
Structural and Allowable ResponsesStructural and Allowable Responses
Allowable peak Allowable peak accelerationacceleration
Peak acceleration at Peak acceleration at corner of the 40th floorcorner of the 40th floor
Torsionally uncoupled systemTorsionally uncoupled system Torsionally coupled systemTorsionally coupled system
22The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
0.2
0.4
0.6
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
.
Normal 常 態 分 佈Lognormal 對 數 常 態 分 佈Type I Extreme Value I型 極 值 分 佈
0.2
0.4
0.6
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
.
Normal 常 態 分 佈Lognormal 對 數 常 態 分 佈Type I Extreme Value I型 極 值 分 佈
Probability of Dissatisfaction with Occupant Comfort CriteriaProbability of Dissatisfaction with Occupant Comfort Criteria
Torsionally uncoupled systemTorsionally uncoupled system Torsionally coupled systemTorsionally coupled system
NormalNormalLognormalLognormal
Type I extreme valueType I extreme value
23The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
ConclusionsConclusions The objective of this paper is to conduct the reliability analysis of high-rise The objective of this paper is to conduct the reliability analysis of high-rise
buildings under wind loads. Two numerical examples, i.e., the torsionally buildings under wind loads. Two numerical examples, i.e., the torsionally uncoupled and coupled systems, are provided to capture the dynamic effects of uncoupled and coupled systems, are provided to capture the dynamic effects of structures with eccentricity between the elastic and mass centers. The structures with eccentricity between the elastic and mass centers. The framework of this research consists of two stagesframework of this research consists of two stages
In the first stage, the occupant comfort criteria are satisfied in the two In the first stage, the occupant comfort criteria are satisfied in the two numerical examples from the viewpoint of deterministic approaches. The peak numerical examples from the viewpoint of deterministic approaches. The peak acceleration of the torsionally coupled system is relatively higher than that of acceleration of the torsionally coupled system is relatively higher than that of the torsionally uncoupled system for each attack angle due to the coupled mode the torsionally uncoupled system for each attack angle due to the coupled mode effectseffects
In the second stage, compared to the lognormal and type I extreme value In the second stage, compared to the lognormal and type I extreme value distributions, the normal distribution can be used to more conservatively distributions, the normal distribution can be used to more conservatively simulate the uncertainties of the eccentricity between the elastic and mass simulate the uncertainties of the eccentricity between the elastic and mass centers in the two numerical examples from the viewpoint of probabilistic centers in the two numerical examples from the viewpoint of probabilistic approaches. The probability of the torsionally coupled system is relatively approaches. The probability of the torsionally coupled system is relatively higher than that of the torsionally uncoupled system for each attack angle due higher than that of the torsionally uncoupled system for each attack angle due to the coupled mode effectsto the coupled mode effects
24The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
Thank You Very MuchThank You Very Much
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