Test Data of High-rise Building Wind Tunnel Tests from Dr. Liangbo

QUAN Yong

In order to study the power spectra and time-spatial correlations of along-wind, across-wind and torsional wind forces loaded on tall buildings with rectangular cross-section, Doctor Liangbo took a series of local wind pressure measurement wind tunnel tests and high frequency balance wind tunnel tests for 6 building models in 1995. In his wind tunnel test report, he showed wind force coefficients of all models, and power spectra, correlations and corresponding mathematical models of generalized wind forces, local wind forces and base forces. In order to build a wind loads database of buildings and structures, I attempt to pick up some useful test data from his report today. 1.Wind Tunnel Test Introduction 1.1. Wind Tunnel The wind tunnel test is carried out in the Boundary-Layer Wind Tunnel with 3.1 meters wide and 2.0 meters high and 16 meters long, Wind Engineering Institute. 1.2. Test Wind Field The blocks and spires are used to generate a boundary layer. The wind tunnel flow features, for example: mean wind speed profile, turbulence intensity and turbulence scale, are shown in Fig.1, According to “AIJ Recommendations for Loads on Buildings”, the experimental flow features belong in terrain roughness category 2.

Fig.1 Features of Testing Wind Field

1.3. Testing Models The length scale of the experimental model was set at 1/500. The dimensions the six testing models, whose cross-section areas are same, were given in the Tab. 1 below.

Tab.1 Model Dimensions

1.4. Measuring Method 1.4.1. Local wind forces

The investigation of local wind forces included the measurement of along-wind force, across-wind force and torsional moment. The levels in the vertical direction are shown in Fig.2.

Fig.2 Measure Levels of Local Forces in the Vertical Direcion

The measured points of each horizontal level were shown from Fig. 3~4.

Fig.3 Measure Points of Local Forces of Each Horizontal Level

Fig.4 Measure Points of Local Wind Forces (D:B=1:1)

Fig. 5 showed the multi-hold method for obtaining wind pressure.

Fig.5 Multi-hold Method for Obtaining Wind Pressure

The time interval of sampling point was 1.6ms, and the total number of sampled points is 16992. So the sampled length is about 27 seconds. The mean and fluctuating wind force coefficients of local wind forces were defined as:

)/()()();/()()();/()()(

)/()()();/()()();/()()(

HMMHFFHFF

HZMHLFHDF

BDqzzCBqzzCBqzzC

BDqzMzCBqzFzCBqzFzC

ZZLLDD

ZLD

σσσ ===′

=== (1)

in which,

)(zFD and )(zFL : mean wind forces at height z; )(z

DFσ and )(zLFσ : standard deviations of wind forces at height z;

)(zM Z and )(zZMσ : mean value and standard deviation of torsional moment at height z;

B, D and qH: projected breadth of model, depth of model and velocity pressure at the top of model. 1.4.2. Base forces The definition of five components of base force was shown in Fig.9.

Fig.9 Definition of five components of base force

The time interval of sampled points is 6.2ms, and the total number of sampled point is 10000. So the sampling length is 62 seconds. The mean and fluctuation wind force coefficients of base forces were defined as:

)/();/();/();/();/(

)/();/();/();/();/(222

222

HMMHMMHMMHFFHFF

HZMHLMHDMHLFHDF

DqBHCDqBHCDqBHCBHqCBHqC

DqBHMCDqBHMCDqBHMCBHqFCBHqFC

ZZLLDDLLDD

ZLDLD

σσσσσ ′=′=′=′=′=

===== (2)

in which, DF and

DFσ : mean value and standard deviation of base shear force in along-wind direction;

LF and LFσ : mean value and standard deviation of base shear force in across-wind direction;

DM and DMσ : mean value and standard deviation of across-wind base overturning moment;

LM and LMσ : mean value and standard deviation of along-wind base overturning moment;

ZM and ZMσ : mean value and standard deviation of torsional moment;

H, B, D and qH: height, breadth, depth of model and velocity pressure at the top of model.

2. Test Results 2.1 Base Force (For Model 1-4) 2.1.1 Wind force coefficients obtained from base forces (For Model 1-4)

Tab.2 Wind force coefficients obtained from base forces

2.1.2 Normalized power spectra of base forces (For Model 1-4)

2.1.3 Cross-correlation between different base forces (For Model 1-4)

2.2 Local Wind Force 2.2.1 Wind force coefficients obtained from local wind force

Tab.3 Wind force coefficients obtained from local wind force

2.2.2 Local wind force coefficients

2.2.3 Normalized power spectra of local wind forces (For Model 1-4)

2.2.4 Auto-correlation coefficient (For Model 1-4)

2.2.5 Root-correlation and phase (For Model 1-4) 2.2.5.1 Same type local wind forces between different levels (For Model 1-4)

2.2.5.2 Different type local wind forces between same level (For Model 1-4)

3. Mathematical Model of Wind Forces 3.1 Root-coherence and Phase 3.1.1 Base forces The root-coherence and phase between across-wind base shear and torsional base moment of models with square cross-section are given by following equations:

HVfBfKfKCOH

/,1)exp(9.0

==

−= (3)

⎪⎩

⎪⎨⎧

≥−−

=<+−−=Φ

00

02

0

)],(50exp[180

1.0,220)2/(16000

ffifff

ffifff (4)

in which,Φ is in degree, f is the frequency (Hz). 3.1.2 Different type local wind forces at the same level The root-coherence and phase between local across-wind force and local torsional moments at level 3H/4 are given by following equations:

HVfBfK

fKCOH

/,5.0

)exp(8.0

==

−= (5)

( )

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

<>

<>+−

≥=−

−=<=−

≥<

<<−−

=Φ

2.01/0

2.01/800160

1/20

)/(01.014.01/160

25.01/60

25.01/480180

00

0

fandBDif

fandBDiff

ffandBDifff

DHffandBDif

fandBDif

fandBDiff

(6)

3.1.3 Same type local wind forces between different levels 3.1.3.1 Local along-wind force The root-coherence and phase between local along-wind forces between level i and level j are given by following equations:

⎪⎩

⎪⎨⎧

>∆−

≤∆−=−=∆∆==

−=

3/,/20.00.1

3/,/11.00.1,,/,6

)exp(

0

0

BDHifHz

BDHifHzRzzzVzffK

fKRCOH

jiH

(7)

0=Φ (8) in which, f is the frequency (Hz), and

ji zz , are heights of level i and j VH is mean wind speed at the top of model 3.1.3.2 Local across-wind force The root-coherence and phase between local along-wind forces between level I and level j are given by following equations:

jiH zzzVzffHzfHzRK

ffifffKR

BDandffifR

BDandffifHzfCOH

−=∆∆=∆=∆−==

⎪⎪⎩

⎪⎪⎨

⎧

≥−−

⎪⎩

⎪⎨⎧

≥<

<<∆−+

=

,/),/(33.0,)/(27.00.1,4

)](exp[

3/

3/)]/(6.01[

00

000

00

0

(9)

jiH zzzVzffHzfHzK

ffifffK

ffifHz

−=∆∆=∆=×∆=Φ=

⎪⎩

⎪⎨⎧

≥−−Φ

<×∆=Φ

,/),/(5.0,180)/(,4

)](exp[

90)/(4.0

00

000

0 (10)

3.1.3.3 Local torsional moment The root-coherence and phase between local torsional moments between level I and level j are given by following equations:

⎩⎨⎧

∆−=≥∆−

=

−=∆∆=∆==

⎪⎩

⎪⎨⎧

≥−−

<=

otherwiseHzBDandBDHifHzR

zzzVzffHzfK

ffifffKR

ffifRCOH

jiH

)/(65.00.11/4/)/(6.094.0

,/),/(33.0,4

)](exp[

0

0

000

00

(11)

[ ]

jiH zzzVzffHzfHzK

ffifffK

ffifHzf

−=∆∆=∆=×∆=Φ=

⎪⎩

⎪⎨⎧

≥−−Φ

<×∆+=Φ

,/),/(23.0,4/90)/(1.6,20

)](exp[

4/90)/(5.120

00

000

0 (12)

3.2 Normalized Power Spectra 3.2.1 Local along-wind force The normalized power spectra of local along-wind forces at any level of building are given by following equations:

( )[ ]

( )( ) ( )( )( ) 5.03.05.0

15.05.010

5.0

12

08.122

)/(142,)/(3.9,//:1/

142,3.9,///:1/

)10/()(,)30/(100

)(1

1

),(

−−−

−

−

===<

===≥

==

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

BDBDBDVLBDif

HzBDVLBDif

zVzVHL

zVfB

Hz

f

ffzfS

HH

HH

H

FD

FD

ηξβ

ηξβ

βη

ξβσ

α (13)

in which, )(zV = mean wind speed at height z 10V = mean wind speed at height 10m α = power law index of vertical profile of mean wind speed HL = turbulence scale at height H 3.2.2 Local across-wind force The normalized power spectra of local across-wind forces at any level of building are given by following equations:

( )[ ]( )

( )[ ]

[ ]

( )

( )

( )

( )⎩⎨⎧

<−

≥=

⎪⎪⎩

⎪⎪⎨

⎧

>=

<<≤+

=

=

==

⎪⎩

⎪⎨

⎧

>=<

=

⎩⎨⎧

>−

≤=

⎩⎨⎧

>

≤=

⎪⎩

⎪⎨⎧

<

≥=

=+

==

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

++

+=

+=

+

−

−

1/16.1

1/6.1

1/63.01//463.0

1/3/146.03/1/0.1/6.1

)3/(4.9

/,/1.0)(

1/)3/()/(21/)3/(1///)3/(4.0

1/)/()1(5.11/5.1

1///0.1

1/0.1

1//5.1/3.1

1///3.1

62.0,)/(38.01

/12.0,/

5.0ln5.0exp2

)(

/48.01

/

/1

/)(

)()()(

3.02

1

1

05.0

04.007.0

3

3.089.02

222

2

72

3

15.212

222

BDifz

BDifB

BDifBDifHB

BDifBDifz

B

HIT

HzzzzzI

BDifHIBDBDifHIBDifBDHI

BDifBDzBDif

c

BDifBD

BDif

BDifBD

BDifBDk

eBDBVffff

fBffS

kf

kfTkf

kfBffS

ffSffSffS

G

zHsss

s

VS

VS

S

Sc

S

S

ST

ST

VS

VS

ST

ST

FL

FL

α

δ

η

ξ

δδ

πδσ

ξσ

σσσ

(14)

in which, )( fSST = effects of the signature turbulence layers )( fSVS = effects of the vortex shedding Sf = Strouhal frequency )(zI = turbulence intensity at height z Gz = gradient height α = power law index of vertical profile of mean wind speed 3.2.3 Local torsional moment The normalized power spectra of local across-wind forces at any level of building are given by following equations:

( )[ ]

[ ]

[ ]{ }

( )( )( )( )( )⎩

⎨⎧

≠+=+

=

⎪⎩

⎪⎨

⎧

>−

=−<+−

=

⎩⎨⎧

>≤

=

⎩⎨⎧

≠=

==

⎩⎨⎧

>≤

=

===⎩⎨⎧

>≤

==

++=

⎩⎨⎧

≥

<===

=+

==

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

+=

++=

−

+

−

−

1/)/(105.01/)/4/(105.0

1/)/(121.01/)/4/(121.01/)/(5.0121.0

1/63.01/2.1

1/0.11/5.1

,75.3

1/)/(5.11/5.1

/,)/(1.0)(,5.1

1/)3/(51/)3/(

,)3/(5.2

))/(1/(sin5.11)(

1/)//(30.01/25.0

,/,/

62.0,)/(38.01

/12.0,/

5.0ln5.0exp2

1)(

5.0ln5.0exp2

1)(

/1

/)(

)(`)(`)()()(

2

21

0

112

05.0

12

13

35.0

3.089.02

2

2

22

22

2

1

21

12

212

2221202

BDifBDzBDifHBz

B

BDifBDzBDifHBzBDifBDz

B

BDifBDif

B

BDifBDif

CCC

BDifBDBDif

HzzzzzIk

BDifHIBDifHI

HI

ffF

BDifBDBDif

kBDVkffff

eBDBVffff

fffS

fffS

kf

kfffS

ffSBffSBffSfFBffS

G

s

mHmmmm

zHsss

s

FP

FP

s

VS

VS

ccS

S

ST

ST

FP

FP

VS

VS

ST

ST

FT

FT

γ

δδ

γπ

ξ

δδ

πδσ

δδ

πδσ

ξσ

σσσσ

α

(15)

in which, )( fSST = effects of the signature turbulence layers )( fSVS = effects of the vortex shedding )( fSFP = effects of the flipping phenomenon Sf = Strouhal frequency )(zI = turbulence intensity at height z Gz = gradient height α = power law index of vertical profile of mean wind speed