Ming-Yi Liu Yu-Jie Li

Ming-Yi LiuMing-Yi LiuYu-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


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


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


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


High-Rise Building ModelHigh-Rise Building Model





Wind load model





Peak factor






The first stage



Design point


The second stage

Reliability index

High-rise High-rise building modelbuilding model


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


Wind Load ModelWind Load Model





Wind load model





Peak factor






The first stage



Design point


The second stage

Reliability index

Wind load modelWind load model


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


DemandDemand





Wind load model





Peak factor






The first stage



Design point


The second stage

Reliability index

DemandDemand


Frequency Domain AnalysisFrequency Domain Analysis



Peak factor





CapacityCapacity





Wind load model





Peak factor






The first stage



Design point


The second stage

Reliability index

CapacityCapacity


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


Reliability AnalysisReliability Analysis





Wind load model





Peak factor






The first stage



Design point


The second stage

Reliability index

Reliability Reliability analysisanalysis


Synthetic MethodSynthetic Method



Design point


Reliability index


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 pointReliability

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


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


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 Fx

ixi(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 Fy

iyi(ω

) (N

2 -sec

/rad

)


10-4

10-2

100

102

104

100

102

104

106

108

1010

1012

ω (rad/sec)

S F

ii (ω

) (N

2 -sec

/rad

)


10-4

10-2

100

102

104

100

102

104

106

108

1010

1012

ω (rad/sec)

S Fx

iyi(ω

) (N

2 -sec

/rad

)


θθθθ

xxxx yyyy

xxyy

Attack Angle = 45˚Attack Angle = 45˚


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

) (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

yi.. yi

..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi.. xi

..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Ryi

.. yi..(

ω)

(m2 /s

ec3 -r

ad)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S R i

.. i..(

ω)

(rad

/sec

3 )

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.. yi

..(ω)

(m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S R i

.. i..(

ω)

(rad

/sec

3 )


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi.. yi

..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi..

i..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Ryi

.. i..(

ω)

(m2 /s

ec3 )


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi..

i..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Ryi

.. i..(

ω)

(m2 /s

ec3 )


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˚


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

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)S R

yi.. yi

..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi.. xi

..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Ryi

.. yi..(

ω)

(m2 /s

ec3 -r

ad)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S R i

.. i..(

ω)

(rad

/sec

3 )


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi.. yi

..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S R i

.. i..(

ω)

(rad

/sec

3 )


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi.. yi

..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi..

i..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Ryi

.. i..(

ω)

(m2 /s

ec3 )


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Rxi..

i..(ω

) (m

2 /sec

3 -rad

)


0 5 10 15 20 25 3010

-20

10-15

10-10

10-5

ω (rad/sec)

S Ryi

.. i..(

ω)

(m2 /s

ec3 )


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


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


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


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


Thank You Very MuchThank You Very Much

Ming-Yi Liu Yu-Jie Li

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