Reliability Analysis of High-Rise Buildings
under Wind Loads
Ming-Yi Liu
Yu-Jie Li
Department of Civil Engineering, Chung Yuan Christian University, Taiwan
Structural Safety Affected by Wind
 Hurricane Katrina, August 2005
 Hyatt Regency New Orleans, USA
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Occupant Comfort Affected by Wind
 Taipei 101, Taiwan, Completed in 2004, 508 m High
 Tuned Mass Damper
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Objectives
 The objective of this paper is to conduct the reliability analysis of
high-rise buildings under wind loads. Numerical examples are
provided to capture the dynamic effects of structures with
eccentricity between the elastic and mass centers. The framework of
this research consists of two stages
 The first stage includes two parts: the deterministic analysis of
wind-induced acceleration for a variety of attack angles, i.e., the
demand, and the determination of allowable acceleration based on
the occupant comfort criteria for wind-excited buildings, i.e., the
capacity
 According to the results obtained in the first stage, the reliability
analysis is conducted in the second stage, which can predict the
probability of dissatisfaction with occupant comfort criteria for a
variety of probability distributions of the structural eccentricity
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Framework
High-rise
building model
Wind load model
Wind load model
Wind velocity profile
High-rise building model
Mass, stiffness and damping matrices
Cross-spectral density function
of wind velocity
Frequency response function
of acceleration
Cross-spectral density function
of wind load
Reliability analysis
Reliability analysis
(synthetic method)
Rackwitz-Fiessler method
Demand
(frequency domain analysis)
Cross-spectral density function
of acceleration
Demand The first stage
Root-mean-square acceleration
at mass center
Design point
Reliability index
Probability of dissatisfaction
with occupant comfort criteria
Root-mean-square acceleration
at corner
Peak factor
The second stage
Finite difference method
Capacity
Capacity
(occupant comfort criteria)
Peak acceleration
at corner
Allowable peak acceleration
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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High-Rise Building Model
High-rise
building model
Wind load model
Wind velocity profile
High-rise building model
Mass, stiffness and damping matrices
Cross-spectral density function
of wind velocity
Frequency response function
of acceleration
Cross-spectral density function
of wind load
Reliability analysis
(synthetic method)
Rackwitz-Fiessler method
Finite difference method
Demand
(frequency domain analysis)
Cross-spectral density function
of acceleration
The first stage
Root-mean-square acceleration
at mass center
The second stage
Design point
Reliability index
Probability of dissatisfaction
with occupant comfort criteria
Root-mean-square acceleration
at corner
Peak factor
Capacity
(occupant comfort criteria)
Peak acceleration
at corner
Allowable peak acceleration
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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N-Story Torsionally Coupled System
Three-dimensional configuration
Elastic center
Elastic
Mass Center Aerodynamic
Center Axis
Center
Axis
Axis
Mass center
Aerodynamic center
Nth Floor
Top view of the ith floor
y
(N-1)th Floor
yi
xi
θi
ith Floor
Bi
Hi
Di
Aerodynamic center
Elastic center
Mass center
Axi
Exi
Bi
ECi
θ
MCi
ACi
Ayi
Eyi
x
2nd Floor
Zi
1st Floor
Di
z
y
θ
x
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Wind Load Model
Wind load model
Wind load model
Wind velocity profile
High-rise building model
Mass, stiffness and damping matrices
Cross-spectral density function
of wind velocity
Frequency response function
of acceleration
Cross-spectral density function
of wind load
Reliability analysis
(synthetic method)
Rackwitz-Fiessler method
Finite difference method
Demand
(frequency domain analysis)
Cross-spectral density function
of acceleration
The first stage
Root-mean-square acceleration
at mass center
The second stage
Design point
Reliability index
Probability of dissatisfaction
with occupant comfort criteria
Root-mean-square acceleration
at corner
Peak factor
Capacity
(occupant comfort criteria)
Peak acceleration
at corner
Allowable peak acceleration
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Wind Load Components
Top view of the ith floor
y
Lift
Lift
Drag
Wind direction
Drag
Wind Direction
ACi
ECi
Bi
θ
TorqueMC
Torque
x
i

Attack angle
Di
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Demand
Wind load model
Wind velocity profile
High-rise building model
Mass, stiffness and damping matrices
Cross-spectral density function
of wind velocity
Frequency response function
of acceleration
Cross-spectral density function
of wind load
Reliability analysis
(synthetic method)
Rackwitz-Fiessler method
Demand
(frequency domain analysis)
Cross-spectral density function
of acceleration
Demand The first stage
Root-mean-square acceleration
at mass center
The second stage
Finite difference method
Design point
Reliability index
Probability of dissatisfaction
with occupant comfort criteria
Root-mean-square acceleration
at corner
Peak factor
Capacity
(occupant comfort criteria)
Peak acceleration
at corner
Allowable peak acceleration
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Frequency Domain Analysis
Demand
(frequency domain analysis)
Cross-spectral density function
of acceleration
Root-mean-square acceleration
at mass center
Root-mean-square acceleration
at corner
Peak factor
Peak acceleration
at corner
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Capacity
Wind load model
Wind velocity profile
High-rise building model
Mass, stiffness and damping matrices
Cross-spectral density function
of wind velocity
Frequency response function
of acceleration
Cross-spectral density function
of wind load
Reliability analysis
(synthetic method)
Rackwitz-Fiessler method
Finite difference method
Demand
(frequency domain analysis)
Cross-spectral density function
of acceleration
The first stage
Root-mean-square acceleration
at mass center
Design point
Reliability index
Probability of dissatisfaction
with occupant comfort criteria
Root-mean-square acceleration
at corner
Peak factor
The second stage
Capacity
Capacity
(occupant comfort criteria)
Peak acceleration
at corner
Allowable peak acceleration
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Occupant Comfort Criteria
Duration of
wind velocity
Occupant
comfort criteria
Return period of
wind velocity
Frequency of
structural oscillation
Melbourne and Palmer (1992)
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Reliability Analysis
Wind load model
Wind velocity profile
High-rise building model
Mass, stiffness and damping matrices
Cross-spectral density function
of wind velocity
Frequency response function
of acceleration
Cross-spectral density function
of wind load
Reliability analysis
Reliability analysis
(synthetic method)
Rackwitz-Fiessler method
Finite difference method
Demand
(frequency domain analysis)
Cross-spectral density function
of acceleration
The first stage
Root-mean-square acceleration
at mass center
The second stage
Design point
Reliability index
Probability of dissatisfaction
with occupant comfort criteria
Root-mean-square acceleration
at corner
Peak factor
Capacity
(occupant comfort criteria)
Peak acceleration
at corner
Allowable peak acceleration
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Synthetic Method
Reliability analysis
(synthetic method)
Rackwitz-Fiessler method
Finite difference method
Design point
Reliability index
Probability of dissatisfaction
with occupant comfort criteria
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Limit State Function and Basic Variables
Original coordinate system
Basic variable Basic variable Z  g  X , X ,, X 
1
2
n
X2
Limit state function
Transformed coordinate system
Basic variable Basic variable Z  g  X 1 , X 2 ,, X n 
X 2
Limit state function
Limit state function
Limit state function
Z 0
Unsafe region
Z 0
Z  0 Unsafe region
Limit state
Z 0
Limit state
Z 0
Safe region
Reliability index Reliability
X1
Basic variable Basic variable
Design point Design point
x1*, x2 *,, xn *
index

Z 0
Safe region
X 1
Basic variable Basic variable
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Numerical Examples
 Two numerical examples, i.e., the torsionally uncoupled and coupled
systems (40-story buildings), are provided to conduct the reliability
analysis of high-rise buildings under wind loads for a variety of
attack angles
 Four types of parameters: the high-rise building model, wind load
model, occupant comfort criteria and reliability analysis, are
considered in this study
 All parameters of the two numerical examples are the same except
the eccentricity between the elastic and mass centers
 Three types of probability distributions: the normal, lognormal and
type I extreme value distributions, are used to model the
uncertainties of the eccentricity between the elastic and mass centers
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Cross-Spectral Density Function of Wind Load
 Attack Angle = 45˚
12
12
10
10
40th Floor
30th Floor
20th Floor
10th Floor
8
10
10
8
10
2
2
S Fxixi(ω) (N -sec/rad)
10
40th Floor
30th Floor
20th Floor
10th Floor
10
S Fyiyi(ω) (N -sec/rad)
10
6
10
4
10
2
10
4
10
2
10
xx
0
10
6
10
yy
0
-4
10
-2
10
0
10
2
10
10
4
-4
10
10
-2
10
ω (rad/sec)
2
10
4
10
ω (rad/sec)
12
12
10
10
40th Floor
30th Floor
20th Floor
10th Floor
8
8
10
2
2
10
40th Floor
30th Floor
20th Floor
10th Floor
10
10
S Fxiyi(ω) (N -sec/rad)
10
10
S Fii (ω) (N -sec/rad)
0
10
6
10
4
10
θθ
2
10
4
10
2
xy
10
0
10
6
10
0
-4
10
-2
10
0
10
ω (rad/sec)
2
10
4
10
10
-4
10
-2
10
0
10
2
10
4
10
ω (rad/sec)
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Cross-Spectral Density Function of Acceleration (1)
 Torsionally Uncoupled System, Attack Angle = 45˚
2
10
xx
10
20
ω (rad/sec)
1
2
-5
25
4
7
5
9
12
11
10
8
-10
2
10
40th Floor
-15
30th Floor
10
20th Floor
10th Floor
The first ten natural
frequency of structure
-15
10
xy
-20
3
10
010
10
15
ω (rad/sec)
515
20
25
4
10
10
20
10
θθ
15
25
20
30
25
10
030
9
10
6
12
2
3
-5
10
-10
40th Floor 40th Floor
30th Floor 30th Floor
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
10
-10
10
-15
10
xθ
-20
030
5
10
0
10
5
15
ω (rad/sec)
10
20
15
25
ω (rad/sec)
10
20
30
25
15
20
25
30
ω (rad/sec)
10
-15
5
ω (rad/sec)
7
3
-5
-20
5
5
1
-5
3
-10
0
0
ω (rad/sec)
10
10
10
30
ω (rad/sec)
2
3
20
30
SRxi..i..(ω) (m /sec -rad)
10
15
25
40th Floor
30th Floor
20th Floor
10th Floor
The first ten natural
frequency of structure
-20
2
515
-20
3
010
10
2
10
-10
10
40th Floor 40th Floor
-15
30th Floor 30th Floor
10
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
-15
yy
-20
3
5
-10
10
2
0
SRxi..yi..(ω) (m /sec -rad)
10
-20
SRxi..i..(ω) (m /sec -rad)
-20
10
10
2
-15
6
5
10
11
10
-10
10
40th Floor 40th Floor
30th Floor 30th Floor
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
-15
10
yθ
-20
30
8
6
-5
3
-10
10
2
10
3
-5
3
SRyi..yi..(ω) (m /sec -rad)
10
40th Floor 40th Floor
30th Floor 30th Floor
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
-15
11
10
3
-10
10
8
-5
10
2
-15
5
-5
SRyi..i..(ω) (m /sec )
12
SRi..i..(ω) (rad/sec )
9
SRyi..yi..(ω) (m /sec -rad)
3
-10
10
2
3
7
10
SRxi..xi..(ω) (m /sec -rad)
10
SRxi..xi..(ω) (m /sec -rad)
4
-5
SRyi..i..(ω) (m /sec )
1
-5
10
-20
0
5
10
10
0
15
5
10
20
ω (rad/sec)
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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25
20
30
25
30
ω (rad/sec)
19
0
Cross-Spectral Density Function of Acceleration (2)
 Torsionally Coupled System, Attack Angle = 45˚
-15
10
ω (rad/sec)
1
2
3
-5
25
4
7
5
9
12
11
10
8
6
3
-10
2
10
40th Floor
-15
30th Floor
10
20th Floor
10th Floor
The first ten natural
frequency of structure
-15
10
xy
-20
10
010
10
15
ω (rad/sec)
515
10
20
20
25
4
10
θθ
20
30
25
10
030
9
10
6
8
12
2
3
1
-5
10
11
40th Floor 40th Floor
30th Floor 30th Floor
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
-10
10
-15
10
xθ
-20
030
5
10
010
515
ω (rad/sec)
10
20
15
25
ω (rad/sec)
10
20
30
25
15
20
25
30
ω (rad/sec)
-10
10
5
ω (rad/sec)
7
5
15
25
10
-15
12
11
3
10
3
2
-5
-20
5
5
1
-5
10
-10
0
0
8
40th Floor
30th Floor
20th Floor
10th Floor
The first ten natural
frequency of structure
-20
ω (rad/sec)
10
10
10
30
-20
ω (rad/sec)
2
3
20
30
SRxi..i..(ω) (m /sec -rad)
10
15
25
10
2
10
20
7
5
-10
3
515
4
10
9
10
40th Floor 40th Floor
-15
30th Floor 30th Floor
10
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
-15
2
010
3
10
-10
2
5
1
2
10
10
yy
-20
SRxi..i..(ω) (m /sec -rad)
0
SRxi..yi..(ω) (m /sec -rad)
10
-20
10
6
5
10
4
11
10
9
7
12
-10
10
40th Floor 40th Floor
30th Floor 30th Floor
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
-15
10
yθ
-20
30
8
6
-5
3
-10
10
xx
-20
6
10
3
-5
SRyi..i..(ω) (m /sec )
10
11
SRyi..i..(ω) (m /sec )
10
40th Floor 40th Floor
30th Floor 30th Floor
20th Floor 20th Floor
10th Floor 10th Floor
The first ten natural
The first ten natural
frequency of structure
frequency of structure
-15
8
2
-10
10
5
-5
SRi..i..(ω) (rad/sec )
10
2
-15
2
3
-5
3
6
12
3
9
10
SRyi..yi..(ω) (m /sec -rad)
-10
10
7
2
3
SRxi..xi..(ω) (m /sec -rad)
10
2
3
SRxi..xi..(ω) (m /sec -rad)
10
4
3
-5
SRyi..yi..(ω) (m /sec -rad)
1
-5
10
-20
0
5
10
010
515
10
20
ω (rad/sec)
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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25
20
30
25
30
ω (rad/sec)
20
Structural and Allowable Responses
Torsionally uncoupled system
90
Torsionally coupled system
Capacity
Demand
90
0.1
120
60
0.08
Allowable peak
acceleration 150
0.06
30
60
0.08
30
0.04
0.02
0.02
180
0
210
330
240
300
270
0.0627
0.06
0.04
Unit: m/sec 2
0.1
120
0.0625
150
Capacity
Demand
180
0
210
Peak acceleration at
corner of the 40th
floor
Unit: m/sec
2
330
240
300
270
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Probability of Dissatisfaction with Occupant Comfort Criteria
Torsionally uncoupled system
90
Torsionally coupled system
.
Normal 常 態 分 佈
Lognormal 對 數 常 態 分 佈
90
1 佈
Type I Extreme Value 型 I極 值 分
120
0.8
1
120
60
0.8
0.6
150
60
0.6
30
150
30
0.4
0.4
0.2
0.2
180
0 180
210
330
240
270
.
Normal 常 態 分 佈
Lognormal 對 數 常 態
Type I Extreme Valu
0
210
Normal
300 Lognormal 240
Type I extreme value
330
300
270
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Conclusions
 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
uncoupled and coupled systems, are provided to capture the dynamic effects of
structures with eccentricity between the elastic and mass centers. The
framework of this research consists of two stages
 In the first stage, the occupant comfort criteria are satisfied in the two
numerical examples from the viewpoint of deterministic approaches. The peak
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
effects
 In the second stage, compared to the lognormal and type I extreme value
distributions, the normal distribution can be used to more conservatively
simulate the uncertainties of the eccentricity between the elastic and mass
centers in the two numerical examples from the viewpoint of probabilistic
approaches. The probability of the torsionally coupled system is relatively
higher than that of the torsionally uncoupled system for each attack angle due
to the coupled mode effects
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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Thank You Very Much
The Fifth International Conference on Reliable Engineering Computing (REC2012), June 13-15, 2012
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