Simplified formulas for evaluation of wind-induced interference effects among three tall buildings

ARTICLE IN PRESS

Journal of Wind Engineering and Industrial Aerodynamics 95 (2007) 31–52 www.elsevier.com/locate/jweia

Simplified formulas for evaluation of wind-induced interference effects among three tall buildings Z.N. Xiea,b, M. Gua, a

State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, People’s Republic of China b Department of Civil Engineering, Shantou University, Shantou 515063, People’s Republic of China Received 21 June 2005; received in revised form 27 April 2006; accepted 4 May 2006 Available online 20 July 2006

Abstract The base-bending moment (BBM) response and the mean BBM of grouped high-rise buildings are studied by a series of wind tunnel tests on typical tall building models using the high-frequency force balance technique. Interference excitations of two upwind buildings with various heights in different upwind terrains are considered. An effective method is proposed to represent the distribution of the envelope interference factor (EIF) among three tall buildings. The results show that two upstream buildings cause more adverse dynamic effects on the downstream building than a single upstream building does. Significant correlations are found in the distributions of the interference factors of different configurations and upwind terrains. Relevant regression equations are proposed to simplify the complexity of the multi-parameter wind-induced mean and dynamic interference effects among three tall buildings. Finally, an example of how to use the data provided in this paper and the proposed methodology is presented. r 2006 Elsevier Ltd. All rights reserved. Keywords: Tall building; Wind tunnel test; Wind load; Base-bending moment; Interference factor

1. Introduction Wind loads on buildings in real environments can be quite different from those measured on isolated buildings in wind tunnels. Surroundings can significantly increase or decrease the wind forces on the interfered buildings. The parameters affecting the wind Corresponding author. Tel./fax: +86 21 65981210.

E-mail address: [email protected] (M. Gu). 0167-6105/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2006.05.003

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forces may include the geometries and orientations of these buildings, the reduced wind velocity (Vr ¼ VH/f0D, where VH is the mean velocity at the top of the building, f0 is the natural frequency of the building, D is the characteristic breadth of the building), number of the adjacent buildings and the upstream terrain condition, etc. Bailey and Kwok [1] investigated the enhanced dynamic response of a tall square building under the interference action from the neighbouring square and circular buildings. Taniike and Inaoka [2] and Taniike [3] investigated the increased response and the possible aeroelastic mechanism of a tall square building under the interference excitation of several types of upstream buildings with different breadths under different upstream flow conditions. The effects of different parameters on the interference effects of tall buildings have also been investigated by many researchers in the past 20 years or so. Some of these studies can be found in [4–9]. However, due to the huge amount of experimental workload, most previous investigations focused mainly on the interference effects between two buildings, that is, one interfering building and one principal building. Only a few studies on the interference effects among three buildings have been reported. Saunders and Melbourne [10] investigated the interference effects on a 150
37
37 m building of upstream single and twin buildings in reduced velocities of 2, 4 and 6. They found that side-by-side arranged two upstream buildings could induce more adverse wind loads on the downstream building than a single upstream building. The interference effects among three or more buildings are still open to research. Xie and Gu [11] made detailed discussions on the mean interference factor (MIF) among three tall buildings. In this paper, the mean interference effects among three tall buildings are further studied to be quantitatively expressed as a simplified diagram. Furthermore, quantitative analyses are also made to determine the effects of variables on the mean and dynamic interference effects among three square tall buildings. In fact, the interference effects among three buildings are very complex and difficult to be expressed in a simple style. In order to reach some critical conclusions that could be used to improve design codes, the envelope interference factor (EIF) is proposed to describe the dynamic interference effects. Some suggestions are finally provided for the assessment of windinduced interference effects on design loads for tall buildings.

2. Description of experiment and analysis 2.1. Experimental equipment Wind tunnel tests were conducted in the STDX-1 Boundary Wind Tunnel of the Department of Civil Engineering at Shantou University. According to the Chinese Load Code (GB50009-2001 [12]), the exposure categories B and D (corresponding to exponents of the power law of mean speed profile of 0.16 and 0.30) are simulated at a length scale of 1/400 by setting spires, barriers and rough elements in the test area. The simulated mean wind profiles (V/Vg), turbulence intensity distributions Iu (%) and the power spectra for the two exposure categories are shown in Fig. 1, where Vg is the mean wind speed at the gradient wind height. The measurements in this paper are carried out by means of the Nitta’s universal forcemoment sensor model No. UFS-4515A100 and the attached signal conditioner and amplifier. The technical specifications for the sensor are shown in Table 1.

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33

Iu (%) 100

0

4

8

12

20

ε V/Vg

90

Terrain B

80

10

0

Measured velocity spectrum at z=210m full scale

Terrain D

70

α =0.16

60

10 -1

α =0.30

nSu(n)/σ2u

z/cm

16

50 40 30

von Karman spectrum

10

nSu ( n)

-2

σ u2

20

=

4nL u / Uz

(1+ 70.8(nLu / U z )2)5 6

with L u=310m in full scale

10 0

0

0.2

0.4

(a)

0.6

0.8

1

V/Vg

10 -3 -3 10 (b)

-2

10

-1

10 nz/ Uz

10

0

10

1

100 Measured velocity spectrum at z=210m full scale

nSu(n)/σu2

10-1

von Karman spectrum nSu ( n) 4nLu / U z = 2 σ u2 1+ 70.8(nL u / Uz )

-2

10

(

)5 6

with L u =290m in full scale

10-3 10 -3 (c)

10-2

10-1 nz/Uz

10 0

10 1

Fig. 1. Simulated wind parameters: (a) wind profiles and turbulence intensity distributions in terrain B and D; (b) longitudinal turbulence spectrum in terrain B; (c) longitudinal turbulence spectrum in terrain D. Table 1 Specifications of Nitta UFS-4515A100 sensor Component

Full scale range

Accuracy

Fx, Fy Fz Mx, My, Mz

440 N 880 N 51 Nm

Linearity: 0.2% FS Hysteresis: 0.2% FS

The lowest natural frequency of the model-balance systems is 112 Hz, which is much higher than the concerned frequency range of the aerodynamic forces acting on the building models. The conditioned and amplified analog signal is transmitted to a Scanivalve’s Zoc/EIM-16 module and eventually digitized by the Scanivalve’s sampling

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platform. Moreover, in order to improve the measurement accuracy of the power spectrum density (PSD) of the aerodynamic base-bending moment (BBM), an additional treatment is made to modify the amplification caused by the model-balance system in the highfrequency band, i.e., SM ðf Þ S M s ðf Þ ¼
s;M
2 ,
H mb ðf Þ


(1)

where S M s;M is the PSD of the measured and amplified ‘‘response’’ of the balance induced by the external aerodynamic
BBM
Ms(t), S M s the PSD of the real signal of the 2 aerodynamic BBM Ms(t) and
H mb ðf Þ
the mechanical admittance of the model-balance system. It is important to distinguish clearly between the amplified BBM, Ms,M(t) and the real BBM, Ms(t). Eq. (1) gives a concise description of the relationship between the real aerodynamic BBM and the amplified BBM. The mechanical admittance of the model-balance system is given as

1
H mb ðf Þ
2 ¼ , (2) 2 2 ð1  ðf =f mb Þ Þ þ ð2zmb f =f mb Þ2 where fmb is the natural frequency of the model-balance system and zmb the critical damping ratio of the model-balance system. In this setup, fmb ¼ 112 Hz and zmb ¼ 0.02. 2.2. Experimental arrangements A 600-mm tall and 100-mm wide square model, made of light foam and skinned with lightweight wood, is used for the principal building. The main parameters and dynamic characteristics of the prototype are 240 m in height, 40 m in breath, 2% for structural damping ratio and 0.2 Hz for natural frequencies for both the sway fundamental modes. Several groups of upstream building models are used as the interfering buildings, which have the same cross section as the principal building but with different heights, which are 0.5, 0.75, 1, 1.25 or 1.5h, where h ¼ 600 mm is the height of the principal building model. All building models are orientated with one face normal to the wind direction and the spacing between them varies as the test parameters in the along-wind direction (x) and the across-wind direction (y) in a grid system as shown in Fig. 2, where A and B are the interfering buildings, and C the principal building at (0, 0). The experimental arrangements for all configurations of the interfering buildings are listed in Table 2, where dx and dy denote the moving steps of the interfering buildings in directions x and y (see Fig. 2). 2.3. Evaluation methods for BBM response The test adopted the high-frequency force balance (HFFB), which is a common technique to study wind load on tall building. The major assumptions made in such a technique are neglecting aeroelastic effects and aerodynamic damping. When the fundamental mode shape of a building is linear or near linear, the PSD and the root mean square (RMS) of the BBM response, S M D ðf Þ and sM D , of a real building can be written as [13]

2 S M D ðf Þ ¼
Hðf Þ
S M S ðf Þ, (3)

ARTICLE IN PRESS Z.N. Xie, M. Gu / J. Wind Eng. Ind. Aerodyn. 95 (2007) 31–52 xA: x-coordinate of model A yA: y-coordinate of model A xB: x-coordinate of model B yB: y-coordinate of model B

-3.2b -2.4b -1.6b

b

Wind

35

-0.8b

B x

C 10.1b

9.1b

8.1b

7.1b

6.1b

5.1b

4.1b

3.1b

2.1b

1.1b

y

0.8b 1.6b 2.4b

A dy

3.2b dx

x y

Fig. 2. x– y-coordinate grid for locating interfering buildings.

Table 2 Experimental arrangements for the interfering buildings Terrain

Number of the interfering buildings

Height of the interfering buildings

Positions of the interfering buildings

Moving step of the interfering buildings

B

1

1.5h 1.25h 1h 0.75h 0.5h

54 54 54 54 54

Small stepa

2

1.5h 1.25h 1h 0.75h 0.5h 1.5h 1.25h 1h 0.75h 0.5h

240 240 2308 240 240 54 54 54 54 54

Large stepb

1.5h 1.25h 1h 0.75h 0.5h

240 240 240 240 240

Large step

D

1

2

a

Small step: dx ¼ 1b and dy ¼ 0.8b. Large step: dx ¼ 2b and dy ¼ 1.6b.

b

Small step Large step Small step

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 w0 SM S ðw0 Þ , jHðf Þj S M S ðf Þ df  sM S 1 þ 4 z0 s2M S 2

sM D ¼ 0

(4)

where wSM S ðwÞ=s2M S denotes the dimensionless PSD of the aerodynamic BBM, SM S ðwÞ can be obtained by Eq. (1), sM S is the standard deviation of Ms(t) and |H(f)|2 is the mechanical admittance of the structure and can be calculated with

1
Hðf Þ
2 ¼ , (5) ð1  ðf =f 0 Þ2 Þ2 þ ð2z0 f =f 0 Þ2 w0 ¼ f 0 D=V H is the reduced natural frequency, D the characteristic breadth of the structure and VH the mean velocity at the top of the structure. The reciprocal of the reduced frequency w0 is the reduced velocity, i.e., Vr ¼

1 VH . ¼ w0 f 0 D

(6)

In fact, Eq. (4) can give the response including background and resonant components of a real building. From Eqs. (4) and (6), one can also find that the BBM response is related to the value of the PSD at the reduced frequency, w0, or in other words, at the reduced velocity Vr. The interference effect on the principal building can be commonly expressed in terms of an interference factor (IF) given by IF ¼

Wind load on a building with interference buildings present , Wind load on an isolated building

(7)

where the wind load can be the mean along-wind base moment (for mean interference effect) or the standard deviation of the BBM response (sMD) in the along-wind or the across-wind direction (for dynamic interference effect). In fact, the interference effects among three buildings are very complex and difficult to be expressed in a simple style. In order to simplify the complexity of the problem and further raise some clauses for building structural design codes, an EIF is proposed here to describe the dynamic interference effects by maximizing the IFs in the reduced velocity ranges of Vr ¼ 2–9, i.e., EIF ¼ max IFðV r Þ, V r 2½2;9

(8)

where the reduced velocities 49 rarely happen for the practical structures [14] and are therefore not considered. Obviously, EIF and MIF are the functions of the positions of the two interfering buildings (see Fig. 2), i.e., EIF ¼ EIFðxA ; yA ; xB ; yB Þ,

(9)

and MIF ¼ MIFðxA ; yA ; xB ; yB Þ.

(10)

Even so, the data analysis of the interference effects is still a complex task. A Windowsbased software platform that integrates spectrum computation, the radial basis functionbased artificial neural network (ANN), and correlation analysis is thus developed to process the test data. The software system can be used to analyze the interference characteristics and mechanism and model the interference effects. With the help of this

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37

software, the IFs at other positions could be predicted by modelling the data and the IF contours could be drawn quickly. 3. Experimental results and discussion 3.1. General characteristics of the EIFs of three identical buildings in exposure category B 3.1.1. Comparison with the two-building configuration and the typical EIFs of three-building configuration Firstly, a comparison between the results of the three-building configuration (with two interfering buildings) and those of the two-building configuration (with only one interference building) is made to illustrate the effects of the number of the interfering buildings. Only the dynamic interference effects are compared here since the dynamic interference effects are more significant and severe than the mean interference effects, and the effects of the number of the interfering buildings on the mean interference effects have been discussed by Xie and Gu [11]. Fig. 3 presents the EIF distributions of the dynamic BBMs on the principal building due to the interference effects caused by an identical interfering building, i.e., two-building configuration, at the various upstream locations in exposure category B. From Fig. 3(a),

-3

2

1. 6

2. 2

2

2.2

9

8

7

6

(a)

2.+4

1 1.4

1.8

5

4

-0.5

1. 6 2

1.6

3

2

C0 0

1

x/b

7

53

2.

-1.5

0.6 1

0.8

1. 4

8

-2

0.4

2 1.

9

-2.5

6

1.6

5

4

3

2

1

y/b

1.8

1.6

1. 8

1.4

2

2

-3

1.6

2.2

2. 2

1

2.4 2

1. 6

2. 4

2

1.4

+ 2 2.4

10 (b)

-1

1.4

1.4

-1.5 0.8

2

1.2

10

2.4

2.2 2.2

2

2

1. 2

1.8

-2.5 -2

1.8

2

1.6

1 1.2

1.8

1.6

y/b

1.6

1.8

1.8

-1 -0.5

C

x/b

Fig. 3. Contours of EIFs of two identical buildings (in exposure category B): (a) along-wind; (b) across-wind.

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significant along-wind interference effects are found when the principal building is located near the high-speed wake boundary of the upstream interfering building. The increased turbulence and mean velocity induced by the upstream building are the key reasons that affect the along-wind dynamic response of the principal building. An EIF as high as 2.42 is measured at x/b ¼ 4.1, y/b ¼ 0.8. Fig. 3(b) shows the distributions of the across-wind envelope dynamic IFs. The significant interference location of the interfering building is found in the region (x/b ¼ 3–10, y/b ¼ 1.8–3.2), where most of the EIFs are greater than 2 and the maximum EIF is 2.42 (the same as the along-wind dynamic interference effects) at x/b ¼ 9.1, y/b ¼ 3.2. Moreover, in the whole position domain of the figure, the most significant across-wind interference effect of the interfering building is found at the critical location of x/b ¼ 0, y/b ¼ 2.4 (i.e., the two buildings are arranged side-by-side and with a spacing of 2.4b) where EIF is 2.53. Another EIF of 2.2 is found when the interfering building is at x/b ¼ 0, y/b ¼ 3.2. This seems to indicate that the side-by-side arranged interfering building can significantly affect the across-wind loads on the principal building. For dynamic interference effects of the three buildings, four position variables (i.e., two x-coordinates and two y-coordinates of the two interfering buildings as shown in Fig. 2) are included in each of the configurations and the results are very difficult to be described with simple contours. A substitute scheme is thus used to analyze the multi-variable test results by fixing one interfering building (model A) at a certain position and varying the spacing between the other interfering building (model B) and the principal building. For the along-wind direction, Fig. 4(a) presents a typical example of the distribution of the EIF when model A is fixed at (3.1b, 2.4b), the critical position of the two-building configuration as shown in Fig. 3(a). It can be found from the comparison results in Figs. 3(a) and 4(a) that the adding model B into the two-building configuration may increase the EIF from 2.2 to about 6.5. For the across-wind direction, Fig. 4(b) illustrates the distribution of the EIFs for model A fixed at (0, 2.4b), which is also a critical position of the two-building configuration, as presented in Fig. 3(b). In general, the interference effects of the three-building configuration are more significant than those of the twobuilding configuration. Particularly, an EIF of 4.55 caused by the two interfering buildings in side-by-side arrangement is also much larger than 2.53 in the two-building configuration, as given in Fig. 3(b). However, Fig. 4 is just a local description of the dynamic interference effects of the three-building configurations and cannot provide the complete information of the interference effects. Statistics analysis for a thorough description of the interference effects is therefore needed and the results are shown in Fig. 5, where p represents the percentage of the positions of the corresponding EIF over the whole test positions of the configurations. From Fig. 5(a), one can see that p is 15% when EIF is about equal to 2.5 for the three-building configuration, but only about 7.5% for the two-building configuration. And the EIFs can still be greater or equal to 3 for 10% of the complete set of interfering building arrangements. This obviously reveals that two interfering buildings can produce stronger interference effects than a single interfering building in the along-wind direction. On the other hand, the results seem to show that the interference effects in the acrosswind direction of two interfering buildings are weaker than a single interfering building for most of the interfering building arrangements, as shown in Fig. 5(b). From this figure, one can see that p is 38.5% when EIF is about 1.0 for the three-building configuration, but only about 15.1% for the two-building configuration. In general, for different levels of

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-3

1.5

2

39

A

1

-2 -1

1

5 5

1.5

3.5

3.5

4.5

3

y/b

1.5

C0

5 6 6.

2

4 .5 4

5.5

3.5

1

2.5

2.5

3

2

2

2

3.5

4

3

3

10

8

6

4

2

0

x/b

2

A 3

1.5

1.5

2

-2

0.5

2

1.5

-1 1

C 0

y/b

(a)

1.5

10 (b)

8

6

1.5

3

3

2

2.5

2 2.5

4

2

2.5 1 2 4 3

1 2 3 0

x/b

Fig. 4. Contours of EIFs of three identical buildings (in exposure category B): (a) along-wind, interfering building model A fixed at (3.1b, 2.4b); (b) across-wind, interfering building model A fixed at (0, 3.2b).

EIF ¼ 1.5, 2 and 2.5, the value of p of two-building configuration is greater than that of three-building configuration. This indicates, generally speaking, that for most cases the dynamic IFs in the across-wind direction of two interfering buildings are smaller than those of one single interfering building. Even so, the EIF is found to be greater or equal to 3 for 1% of the whole sets of interfering building arrangements. This 1% of arrangements, corresponding to about 22 groups of building arrangements including the side-by-side arrangement, can produce more adverse dynamic wind loads on the principal building. The dynamic response, particularly in the along-wind direction, is sensitive to the turbulence structure of approaching flow, especially to the vortices shedding from upstream buildings. Fig. 4(a) illustrates that the two upstream buildings in a staggered arrangement do produce stronger fluctuating flow on the principal building. Comparison of PSDs of the BBM of the principal building with and without interference may explain the mechanism of the interference to some extent. Fig. 6 shows the comparison of the

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40

50 Two buildings Three buildings

41.5%

40

36% 36% 29%

p/%

30 20 10

15%

15%

10%

7.5%

6% 2%

0 0.5

1

1.5

2

(a)

2.5 EIF

3

2%

3.5

4

4.5

50 Two buildings Three buildings

41.5%

40

38.5%

36.5%

34%

p/%

30 20.4%

20

15.1% 9.4%

10

3.6%

0 0.5 (b)

1

1.5

2 EIF

2.5

1%

3

3.5

Fig. 5. Comparison of distribution of IFs between two- and three-building configurations in exposure category B (a) along-wind, (b) across-wind.

non-dimensional PSD of the along-wind aerodynamic BBM of the principal building without the presence of the interfering buildings and with the interfering buildings at (3.1b, 2.4b) and (5.1b, 0.8b). The non-dimensional PSD, which is expressed by fS My ðf Þ=q2H , where qH ¼ 12rV 2H Dh2 denotes the reference moment, is the function of the reduced frequency, fD/VH. From Fig. 6, one can see that the PSD of the aerodynamic BBM of the principal building is significantly interfered by the two interfering buildings. The amplitudes of the PSD of the principal building under interference are larger than that of the alone principal building at all of the frequencies, and the dominant frequencies in the approach flow caused by vortex shedding from the two interfering buildings are registered on the PSD in the peak at the reduced frequency of about 0.1. As a result, this leads to a high EIF of about 6.5, as shown in Fig. 4(a). 3.1.2. Simplification of results for three-building in arbitrary configurations The above discussions illustrate the basic characteristics of dynamic interference effects among three buildings. Since the four variables, i.e., two x-coordinates and two ycoordinates of the two interfering buildings, are involved in the analysis of the interference effects of three-building configurations, the EIFs cannot be simply expressed in a single

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41

10 -1

f S My(f ) /qH2

10 -2

10 -3

10 -4

10 -5

10 -6 -3 10

Isolated building Interfering buildings at (3.1b,-2.4b) and (5.1b,0.8b)

10

-2

10

-1

0

10

f D/VH Fig. 6. Comparison of non-dimensional PSD of along-wind aerodynamic BBM of principal building with and without presence of interfering buildings.

contour as in the two-building cases. The problem is how to propose a simple but precise enough representation method for practical applications from the complex data. Based on the ANN modelling of distribution of the EIFs, a new simplified contour that can clearly illustrate the critical positions of the two interfering buildings is proposed by synthesizing the effects over the whole test domain. The results are presented in the following. Firstly, the ANN method is used to model the distribution of the EIF, i.e., create the relation between the EIF and the positions of the two interfering buildings A and B by training the ANN model, using the four position variables (i.e., two x-coordinates and two y-coordinates of the two interfering buildings) as the input parameters. About 2200 groups of test data are used to train the neural network for the configurations of three identical buildings in the along-wind and across-wind directions, respectively. Once the neural network, Net, is trained, it can be used to predict more EIFs that have not been tested by EIF ¼ NetðPA ; PB Þ;

PA ðx; yÞ; PB ðx; yÞ 2 O,

(11)

where O denotes the whole position domain of the interfering buildings in the test shown in Fig. 2. With the trained Net, more EIFs at the other positions of the test domain O can be predicted. The results for the along-wind and across-wind directions of the configurations of three identical buildings are presented in Fig. 7. In Fig. 7, SAB denotes the spacing between the two interfering buildings; A, A1 and A2 denote the critical regions of the interfering building model A; B, B1, B2 and B3 denote the critical regions of the interfering building model B. Fig. 7 clearly indicates that significant interference effects will occur when the two interfering buildings are both located at their critical regions. For example, when the two interfering buildings are located at about (0, 3b) and (0, 3b), they will induce significant interference effect in the across-wind direction on the principal building with an EIF of larger than or equal to 4.0. And, when

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42

-3 P (9b,-2b)

A

B

-2

EIF=5.0

-1 SAB

PA (4b,-b)

y/b

EIF ⭓ 4.3,SAB >3b EIF ⭓3.8,SAB >2.7b

C

EIF ⭓ 3.5,SAB >2.5b

P (4b,b) A

EIF ⭓ 3.0,SAB >2.2b

1 B

2 3

PB (9b,2b)

10

9

8

7

5 x/b

6

(a)

4

2

3

1

0

A2

-3

A1

-2

? EIF=4.0

-1

α

B2

EIF ⭓2.6 EIF ⭓2.4

y/b

EIF ⭓ 3.4

C

SAB ⭓ 2.5b α ∈ [30° ,150° ]

SAB

1 2

B3

10 (b)

9

8

7

6

5 x/b

B1

4

3

2

1

3 0

Fig. 7. Critical positions of two interfering buildings and EIFs for three identical building configurations (in exposure category B): (a) along-wind, (b) across-wind.

the two interfering buildings are at about (3b, 2b) and (4b, 1b), the EIF in the along-wind direction will be X4.3 and o5. The above expressions simplify the complexity of the problem and can help the designer judge quickly, to a good degree of approximation, the extent and the severity of the dynamic interference effects among three tall buildings.

3.2. Effects of height ratio To investigate the effects of the height ratio (hereafter referred to as Hr) of the interfering buildings to the principal building on the interference, five pairs of interfering building models which have the same breadth as the principal building but different heights are adopted for the test. The height ratios are 0.5, 0.75, 1.0, 1.25 and 1.5. In order to quantify the effect of Hr, the factors of the four configurations of Hr ¼ 0.5, 0.75, 1.25 and 1.5 are compared with those of the configuration of Hr ¼ 1. The results from the test in exposure category B are discussed in the following.

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3.2.1. Mean interference effect Generally, the mean interference effects of tall buildings present ‘‘shielding effects’’, that is to say, the presence of existing nearby interfering buildings tends to decrease the mean wind load on the principal building. The authors have already given basic characteristics of the MIF among three tall buildings [11]. Effects of the height ratio on the MIF are further discussed in this section by correlation analysis. The results show that the MIFs (xA/b, yA/b, xB/b, yB/b) of the four configurations of Hr ¼ 0.5, 0.75, 1.25 and 1.5 have pronounced correlations with those of the configuration of Hr ¼ 1. Fig. 8 presents a comparison of the MIFs between the configurations of Hr ¼ 1.5 and Hr ¼ 1.0. In Fig. 8, the data are regressed as linear expression since the stronger correlations are found between the two sets of the MIFs; RIF is the linear regression interference factor; e ¼ 0.048 is the residual that denotes the accuracy of the regression; and r ¼ 0.99 is the correlation coefficient that denotes the degree of correlation. The closer the value of the correlation coefficient is to 1, the better linear correlation between the two sets of the test MIFs data. The regression curves of the mean RIF for the configurations of different height ratios are shown in Fig. 9, which indicates that the shielding effects of the interfering building decrease with the decrease of Hr and become negligible when Hro0.5. The IF decreases, or in other words, the shielding effect increases, with the increase of Hr in the range between 0.5 and 1.0. In contrast to the significant difference of the MIFs for Hr ¼ 0.5, 0.75 and 1.0, the IFs vary slightly for HrX1, and the two regression curves of HrX1.25 are almost the same (see Fig. 9). This means that the shielding effects keep unchanged for HrX1.25. The effects of Hr on the MIF among three buildings shown in Fig. 9 are quite similar to those of two-building configuration [15].

1.5 Tested RIF=-0.096 +1.106 MIF

MIF for Hr = 1.5

1

0.5

0

-0.5 -0.5

0

0.5

1

1.5

MIF for Hr = 1 Fig. 8. Correlation analysis of MIFs between configurations of Hr ¼ 1 and Hr ¼ 1.5 (e ¼ 0.048, r ¼ 0.99).

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1.5

RIF

1.0

0.5 Hr =0.50,ε=0.023,ρ=0.94 Hr =0.75,ε=0.043,ρ=0.97

0.0

Hr =1.00,ε=0,ρ=1.0 Hr =1.25,ε=0.037,ρ=0.99 Hr =1.50,ε=0.048,ρ=0.99

-0.5 -0.5

0.0

0.5

1.0

1.5

MIF for Hr = 1.0 Fig. 9. Regression results of MIF for configuration of different height ratios (in exposure category B).

Based on a large quantity of computation, the MIF for the different height ratios can be simply predicted by

RIF ¼

8 0:799 þ 0:181MIF; > > > < 0:413 þ 0:58MIF;

H r ¼ 0:5; H r ¼ 0:75;

> MIF; > > : 0:096 þ 1:106MIF;

H r ¼ 1:0; H r X1:25:

(12)

3.2.2. Dynamic interference effect Consider the effects of the height ratio on the dynamic IFs, the EIFs between different height ratios still show good correlations. An example of the comparison between the EIFs of the configuration of Hr ¼ 1.5 and 1.0 is shown in Fig. 10, where the data are also regressed as linear expression. The other three configurations of Hr ¼ 0.5, 0.75 and 1.25 also have good linear correlations with that of the configuration of Hr ¼ 1. Fig. 11 presents the regression results of the EIFs for the different height ratios. Generally, the dynamic interference effects increase with the height of the interfering building, while the effects for Hro0.5 can be neglected. For the along-wind direction, the EIFs increase rapidly with the increase of Hr in the range between 0.5 and 1.25. However, in contrast to the significant difference of the EIFs of the interfering buildings with Hr ¼ 0.5, 0.75 and 1.0, the EIFs in the across-wind direction vary slightly for the interfering buildings of HrX1, and the three regression curves of HrX1 are very close, as shown in Fig. 11(b). Based on a great quantity of computation, the regression relations of the EIFs for the different configuration of Hr can be summarized for the along-wind and across-wind

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10 Tested RIF=-1.524+2.256 EIF

EIF for Hr =1.5

8

6

4

2

0 0

1

2

3

4

5

4

5

EIF for Hr =1

(a) 6

Tested RIF=-0.067+1.118 EIF

5

EIF for Hr =1.5

4

3 2

1

0

0

1

(b)

2

3

EIF for Hr =1.0

Fig. 10. Correlation analysis of EIFs between configurations of Hr ¼ 1 and Hr ¼ 1.5: (a) along-wind,
¼ 0:76, r ¼ 0:85 and (b) across-wind,
¼ 0:34, r ¼ 0:83.

directions, respectively, as 8 1:011 þ 0:031EIF; > > > > > > < 0:698 þ 0:408EIF; RIF ¼ EIF; > > > 1:317 þ 1:988EIF; > > > : 1:524 þ 2:256EIF;

H r ¼ 0:5; H r ¼ 0:75; H r ¼ 1; H r ¼ 1:25; H r ¼ 1:5;

(13)

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10 Hr =1.50 Hr =1.25

8

Hr =1.00

6 RIF

Hr = 0 .75 Hr =0.50

4

2

0

1

2

(a)

3

4

5

EIF 5

Hr=1.50 Hr=1.25

4

Hr= 1 .00 Hr= 0 .75

3 RIF

Hr=0.50

2

1

0

0

1

(b)

2

3

4

5

EIF

Fig. 11. Regression results of EIF for configuration of different height ratios (in exposure category B) (a) alongwind, (b) across-wind.

and 8 1:018 þ 0:068EIF; > > > > > > < 0:627 þ 0:45EIF; RIF ¼ EIF; > > > 0:028 þ 1:038EIF; > > > : 0:067 þ 1:118EIF;

H r ¼ 0:5; H r ¼ 0:75; H r ¼ 1; H r ¼ 1:25;

(14)

H r ¼ 1:5:

From Eqs. (13) and (14), one can predict the EIFs of Hr6¼1 from the EIFs of three identical building configurations shown in Fig. 7.

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3.3. Effect of upstream terrain 3.3.1. Mean interference effect For the effects of the upstream terrains on the MIF of the configuration of different Hr, strong linear correlations exist in the MIFs between the two upwind terrains for all the configurations of height ratio, which can be seen in Fig. 12. From Fig. 12, one can see that the MIF of any configuration of Hr in exposure category D can be simply predicted from the corresponding MIF in exposure category B by MIFD ¼ 0:078 þ 0:982MIFB .

(15)

3.3.2. Dynamic interference effect The EIFs of the configuration of three identical buildings between exposure categories B and D are compared in Fig. 13, where EIFB and EIFD are the EIF in exposure categories B and D, respectively. It can be found that the data also show a good correlation. Similar to the configuration of three identical buildings, correlations still exist in the EIFs of the other height ratios between the two categories of terrains. Then, the relation of the EIFs between the two upstream terrains for all the height ratio configurations can be expressed by Eqs. (16) and (17) for the along-wind and across-wind directions, respectively: 8 0:946 þ 0:093EIFB ; > > > > > > < 0:819 þ 0:181EIFB ; EIFD ¼ 0:599 þ 0:332EIFB ; > > > 0:806 þ 0:273EIFB ; > > > : 0:757 þ 0:314EIFB ;

H r ¼ 0:5; H r ¼ 0:75; H r ¼ 1; H r ¼ 1:25;

(16)

H r ¼ 1:5;

1.5

MIFD

1

0.5

Hr =0.5 Hr =0.75 Hr =1.0

0

Hr =1.25 Hr =1.50

-0.5 -0.5

0

0.5 MIFB

1

1.5

Fig. 12. Correlations of MIFs between terrain categories B and D.

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2.2 Tested RIF=0.62+0.317 EIF

2 1.8

EIFD

1.6 1.4 1.2 1 0.8 0.6 0

1

2

(a)

3

4

5

3

4

5

EIFB Tested RIF= 0.227+0.736 EIF-0.084 EIF2

1.8 1.6

EIFD

1.4 1.2 1 0.8 0.6

0

1

(b)

2 EIFB

Fig. 13. Correlations of EIFs between different upwind terrains for configuration of three identical buildings: (a) along-wind and (b) across-wind.

8 0:853 þ 0:103EIFB ; > > > > > 0:725 þ 0:255EIFB ; > < 0:227 þ 0:736EIFB  0:084EIF2B ; EIFD ¼ > 2 > > > 0:072 þ 0:951EIFB  0:238EIFB ; > > : 0:059 þ 0:97EIF  0:132EIF2 ; B

B

H r ¼ 0:5; H r ¼ 0:75; H r ¼ 1;

(17)

H r ¼ 1:25; H r ¼ 1:5:

The relations between EIFB and EIFD in the across-wind direction for the height ratio of HrX1 are expressed by a second-order regression polynomial rather than a linear formula. Fig. 14 illustrates the variations of the EIF for the configurations of different height ratios in the exposure categories D with those in category B.

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3.5 H r=0.5 H r=0.75

3

H r=1.0 H r=1.25

2.5 EIFD

H r=1.5

2

1.5

1

0.5 0

2

4

6

8

EIFB

(a) 2 H r=0.5

1.8

H r =0.75 H r =1.0

1.6

H r =1.25 H r =1.5

EIFD

1.4 1.2 1 0.8 0.6

0

(b)

1

2

3

4

5

EIFB

Fig. 14. Regression results of EIFs between different upwind terrains for configuration of different height ratios: (a) along-wind and (b) across-wind.

4. Application considerations Here presents a discussion on how to use the above results in design of real tall buildings. For design of a real structure, the required peak loads can be obtained using, for example, the BBM-based moment gust loading factor (MGLF) suggested by Kareem and Zhou [16]. The MGLF for an isolated building is defined as [16]: ^ G M ¼ M=M,

(18)

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^ the peak BBM response. The MGLF where GM is the MGLF, M the mean BBM and M can be computed by sM D , (19) M in which gM is the peak factor, and sM D the RMS of the BBM response as defined in Eq. (4). For a building when adjacent buildings are present, the MGLF can be defined as G M ¼ 1 þ gM

^ I =M, G M;I ¼ M

(20)

where GM,I is the MGLF of the interfered principal building, M the mean BBM for the ^ I the peak BBM response of the principal building. According to isolated building and M ^ the above definition, M I can thus be expressed by ^ I ¼ M I þ gM sM D;I ¼ MIF
M þ EIF
gM sM D M ¼ MIF
M þ EIF
ðG M  1Þ
M ¼ ðMIF þ EIF
ðG M  1ÞÞ
M,

ð21Þ

where M I and sM D;I are the mean BBM and the RMS of the BBM response of the principal building, respectively. Substituting Eq. (21) into (20) leads to G M;I ¼ MIF þ EIF
ðGM  1Þ,

(22)

where MIF is the mean interference factor, EIF the envelope dynamic interference factor and GM, defined in Eq. (18), is the MGLF of the isolated building. To further explain the application of the above method, a simple example is given here. For an isolated tall building, the MGLF is assumed to be GM ¼ 1.4 in terrain B. When the building is aerodynamically interfered by two adjacent identical buildings A and B at PA(4b, b) and PB(9b, 2b), the MGLF will be computed with the above test results and the present method. The MIF of the interfered building is found to be 0.81 according to the reduced IF contour proposed by Xie and Gu [11], i.e., MIF ¼ 0.81. For the dynamic interference effect, considering the symmetrical positions of PA and PB in Fig. 7(a), it can be seen that 3:0pEIFðP0B ; P0A Þp3:5.

(23)

For practical purpose, EIF could be conservatively estimated by EIF ¼ EIFðPA ; PB Þ ¼ EIFðP0B ; P0A Þ ¼ 3:5.

(24)

Finally, the MGLF of the building under the interference could be computed. G M;I ¼ MIF þ EIF
ðGM  1Þ ¼ 0:81 þ 3:5ð1:4  1Þ ¼ 2:21.

(25)

5. Conclusions Effects of the upstream terrain conditions, the relative heights of the interfering buildings, and the spacing between two and three buildings on the mean and dynamic interference factors are investigated by a series of deliberate wind tunnel tests and detailed

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analyses using correlation and regression methods in this study. Some of the main results are summarized below: 1) It is highlighted in this study that the two upstream buildings can cause more adverse dynamic effects on the principal building than a single upstream building does. An effective method is proposed to represent the critical distribution of the envelope interference factor (EIF) of the interference effects among three tall buildings. 2) Two interfering buildings can produce stronger along-wind dynamic interference effects than a single interfering building, but the dynamic interference effects in the across-wind direction caused by two interfering buildings seem to be somewhat weaker than those by a single interfering building for most of the interfering building arrangements. 3) Higher interfering building produces stronger dynamic interference effects on the principal building whilst the effect of the interfering building with Hro0.5 can be neglected. 4) Significant correlations exist in the mean and dynamic interference factors for different configurations of height ratios and upwind terrains. Regression equations reflecting the inherent complex relationships are proposed to simplify the expressions of the interference effects. Finally, it should be pointed out that the interference effects for different breadths between the interfering building and the principal building were also carried out in this research, but unfortunately, the results show an obvious divergence and no correlation among the EIFs for the different configurations of breadth ratios. More tests and efforts are therefore needed. Acknowledgments This research is jointly supported by the National Science Foundation (50478118, 50321003), the Foundation for University Key Teachers by the Ministry of Education and the Science Foundation of Guangdong Province (010455). Their supports are gratefully acknowledged. References [1] P.A. Bailey, K.C.S. Kwok, Interference excitation of twin tall buildings, J. Wind Eng. Ind. Aerodyn. 21 (1985) 323–338. [2] Y. Taniike, H. Inoka, Aeroelastic behaviour of a tall building in wakes, J. Wind Eng. Ind. Aerodyn. 28 (1998) 317–327. [3] Y. Taniike, Turbulence effect on mutual interference of buildings, J. Eng. Mech. ASCE 117 (3) (1991) 443–456. [4] A.C. Khanduri, T. Stathopoulos, C. Be´dard, Wind-induced interference effects on buildings—a review of the state-of-the-art, Eng. Struct. 20 (7) (1998) 617–630. [5] A.C. Khanduri, C. Bedard, T. Stathopoulos, Modelling wind-induced interference effects using backpropagation neural networks, J. Wind Eng. Ind. Aerodyn. 72 (1997) 71–79. [6] H. Sakamoto, H. Haniu, Aerodynamic forces acting on two square prisms placed vertically in a turbulent boundary layer, J. Wind Eng. Ind. Aerodyn. 31 (1988) 41–66. [7] A. Thepmongkorn, G.S. Wood, K.C.S. Kwok, Interference effects on wind-induced coupled motion of a tall building, J. Wind Eng. Ind. Aerodyn. 90 (2002) 1807–1815.

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[8] U.F. Tang, K.C.S. Kwok, Interference excitation mechanisms on a 3DOF aeroelastic CAARC building model, J. Wind Eng. Ind. Aerodyn 92 (2004) 1299–1314. [9] P. Huang, M. Gu, Experimental study on wind-induced dynamic interference effects between two tall buildings, Wind Struct. Int. J. 8 (3) (2005) 147–161. [10] J.W. Saunders, W.H. Melbourne, Buffeting effects of upwind buildings, in: Proceedings of the Fifth International Conference on Wind Engineering, Fort Collins, CO, 1979, Pergamon Press, Oxford, 1980, pp. 593–605. [11] Z.N. Xie, M. Gu, Mean interference effects among tall buildings, Eng. Struct. 26 (2004) 1173–1183. [12] GB50009-2001, Chinese Load code for design of building structures, 2002. [13] T. Tschanz, A.G. Davenport, The base balance technique for the determination of dynamic wind loads, J. Wind Eng. Ind. Aerodyn. 13 (1983) 429–439. [14] Z.N. Xie, Wind-induced interference effects on typical tall buildings, Ph.D. Thesis, Tongji University, China, 2003. [15] Z.N. Xie, M. Gu, A correlation-based analysis on wind-induced interference effects between two tall buildings, Wind Struct. Int. J. 8 (3) (2005) 163–178. [16] A. Kareem, Y. Zhou, Gust loading factor—past, present and future, J. Wind Eng. Ind. Aerodyn. 91 (2003) 1301–1328.