Effects of Corner Modification on the Wind-Induced Responses of High-Rise Buildings

Abstract

Aerodynamic optimization of building geometry has received significant attention in the design community. In this paper, a process with the high-frequency force balance (HFFB) technique to determine the most effective mitigation measure and the synchronized pressure integration (SPI) technique to verify the effect is developed for the aerodynamic optimization of high-rise buildings. Then, the process is applied to a 318 m-tall high-rise building. Tests show that the wind force on the building will not be symmetrical about the wind azimuth due to the interfering effect. The standard deviation of the base bending moment in the cross-wind direction is much larger than that in the along-wind direction. It indicates that the cross-wind loads will be dominated, providing a remarkable building height. The aerodynamic treatment of corner modifications has a considerable benefit in reducing the cross-wind loads and responses. Among the four corner modifications, the model with a 10% roundness radius to width ratio has the best mitigation effect in the along wind and cross-wind direction. Furthermore, the mean and extreme base overturning moments obtained by the SPI and the HFFB tests almost coincided with wind azimuth with acceptable discrepancy.

1. Introduction

With the burgeoning growth of the economy and construction industry, high-rise buildings are being widely built to meet the demand for social development [1]. Wind loads are one of the most critical parameters in the design stage to reduce the occupants’ potential discomfort and the structure’s cost. Due to the aerodynamic features, such as flow separation, wind effects on high-rise buildings are intrinsically sensitive to their external shapes [2]. Therefore, aerodynamic optimizations of building geometry have received significant attention in the design community. Typical approaches to aerodynamic optimization could be divided into two categories: minor modifications in the plan size, such as corner recessing, roundness, and chamfering, and the other is major modifications in the elevation size, such as tapering, setback, and twisting [3]. These approaches could reduce the cross-wind responses originating from the alternate vortex shedding from both sides of the building and intensified by the high-rise building’s lower frequencies and damping levels. However, the major modifications have several inherent drawbacks, such as insufficient building area and excessively high construction costs [4,5]. In contrast, minor modifications can maintain the original form to the utmost degree and coordinate with the surrounding space. The minor modifications in the cross-section can lead to significant changes in the airflow mechanism, resulting in the reduction in wind load acting on the buildings.

The typical corner modifications mainly contain roundness, chamfering, and recession. At present, several studies have been conducted via various techniques, including the high-frequency force balance (HFFB), synchronized pressure integration (SPI), aeroelastic (AE) test, and computational fluid dynamics (CFD) to investigate the influence of section modifications on the wind-induced response of high-rise buildings.

Quan et al. [6] carried out force measurement tests on 15 building models in four terrain exposures by using the HFFB technique. They investigated the aerodynamic spectra and base force coefficients with different chamfering and recession rates. They confirmed that a modification ratio of around 10% would mitigate the amplitude of the aerodynamic spectrum in most frequency ranges. Tse et al. [7] combined the aerodynamic optimization measures with the economic benefits of high-rise buildings via the HFFB technique. The aerodynamic spectrum and wind-induced response of structures with different chamfering and recession rates under the same total building volume and different heights are provided. They concluded that the recession measure has a better effect in reducing the along wind and cross-wind base overturning moment than the chamfer measure. Zhang et al. [8,9] used the HFFB technique to investigate the base forces of high-rise buildings after corner recession, chamfering, and roundness. They examined the influence of the modification rate on the base aerodynamic forces and fitted the empirical formula of base force coefficient with the varied parameters.

Irwin et al. [10] applied the corner modification method to the 509 m Taipei 101 building through the SPI test. They found that after the section with a double recession, the base overturning moments of the structure would be scaled down by 25%. Xie et al. [11] conducted wind tunnel tests on super high-rise buildings with the taper along the height by the SPI method. They found that for the high-rise building with a taper ratio of 2.2%, the peak value of the base overturning moment after corner modification is 31% lower than that without chamfering. Li et al. [2] conducted SPI tests on four different section types of buildings, i.e., square, chamfering, recession, and roundness. They obtained the base moment coefficients, local wind coefficients, and aerodynamic spectra of the structures. They found that the recession and chamfering measures are more beneficial in alleviating the wind load on the buildings than the others.

Kawai et al. [12] studied the influence of recession, chamfering, and roundness on the aerodynamic characteristics of high-rise buildings via the aeroelastic wind tunnel test. They obtained the base forces and power spectra in the along wind and cross-wind directions under different wind speeds. They found that a minor degree of chamfering apparently reduces the base forces compared with the original square section. Cao et al. [13] identified the aerodynamic damping ratio via aeroelastic tests on ten high-rise buildings and explored the influence of chamfering and recession on the ratio. They stated that the aerodynamic damping ratio increased with the increase in the modification ratio. Wang et al. [14] studied the influence of recession and roundness rates on the wind-induced vibration responses through aeroelastic tests and obtained the cross-wind displacement atop the building. They declared that the recession rates in the range of 7.5–10% could significantly attenuate the displacements around critical wind speed but increase the St of buildings, leading to the vortex-induced vibration of structures at a lower wind speed.

Liao et al. [15] conducted CFD simulations on the building with 10% chamfering and recession measures on the section. They found that chamfering actions would better mitigate the drag coefficients than the recession measure. Elshaer et al. [16] combined a genetic algorithm and an artificial neural network to study the optimal section shape of different corner parameters such as the convex and concave degrees. Ding et al. [17] used the multi-assurance proxy model to give the optimal solution for various cross-section profile parameters. In their work, a large number of CFD data proxy models with fast calculation under different section conditions were utilized to find the optimal solution with the optimization algorithm. Yang [18] used the agent model combined with the optimization method of NSGA-2 to study the wind effect of convexity and concavity of building sections on wind-induced response. He found that the convexity of the building makes the building shape more streamlined and suppresses the intensity of vortex shedding. Abdelaziz et al. [19] presented a data-driven adaptive control strategy that continuously seeks to minimize wind-induced vibration for a given average flow condition via CFD technique. However, the CFD simulation is closely related to the turbulence model in the pretreatment process. Thus, its accuracy still requires further confirmation by the physical meanings, such as the wind tunnel test [20].

The above studies are shown in

Table 1. Summaries of the studies on corner modifications of high-rise building.

Reference	Corner Modification	Method	Parameters Involved
			Pressure Coefficient	Base Force	Aerodynamic Spectrum	Wind-Induced Response
Quan et al. [6]	CH, RE, RO	HFFB		√	√
K.T. Tse et al. [7]	CH, RE	HFFB			√	√
Zhang et al. [8,9]	CH, RO, RE	HFFB		√
Irwin et al. [10]	RE	SPI		√		√
Xie et al. [11]	RE	SPI		√	√	√
Li et al. [2]	CH, RO, RE	SPI	√	√	√
Kawai [12]	CH, RE, RO	AE				√
Cao et al. [13]	CH, RE	AE				√
Wang et al. [14]	CH, RO, RE	AE				√
Liao et al. [15]	CH, RE	CFD	√	√
Elshaer et al. [16]	RE	CFD	√		√	√
Ding et al. [17]	RO, CH	CFD	√
Yang [18]	RO	CFD	√
Abdelaziz [19]	RE	CFD	√	√	√

, in which the roundness, chamfering, and recession are denoted as “RO”, “CH”, and “RE”, respectively, for the purpose of simplicity.

Most studies mentioned above primarily focused on corner modifications, and there is no definite answer to the optimal solution for corner modifications. The optimal aerodynamic modification of a high-rise building may vary due to the section shape and surrounding buildings, etc. During the design stage of an actual building, a specific analysis addressing the aerodynamic optimization issue may be necessary providing a high-rise building with remarkable height. In the actual engineering projects, except for the building shape, the local factors, such as nearby interference buildings, terrain conditions, and design wind speed, will also have additional effects on the wind load and wind-induced response of the high-rise buildings, and therefore the determination of the aerodynamic modification measure. Furthermore, for a high-rise building with remarkable height, a SPI test on the building may also be applied to determine the wind loads on the building, components, and cladding; therefore, the comparison between the results of the HFFB and SPI tests can be completed so as to verify the effect of aerodynamic modification measure.

In this paper, a process with the combination of the HFFB technique to determine the most effective mitigation measures and the SPI technique to verify the effect is developed to provide an operable scheme for the aerodynamic optimizations of an actual high-rise building. A 318 m-tall high-rise building in Zhejiang, China is taken as an example to illustrate the process. First, two aerodynamic modification measures, including corner roundness and recession with different rates, were investigated through HFFB tests. Then, the wind-induced displacement, acceleration, and base overturning moment are obtained after considering the dynamic structural properties of the building, and the corner modification with the best mitigation effect is determined. Finally, the impact of the determined corner modification is verified by the SPI test, with which the results of the HFFB test are compared. The developed process and results presented herein are helpful and meaningful for the wind-resistant design of actual high-rise buildings.

2. Experiment Setup

A high-rise building in Zhejiang, China is investigated to illustrate the process, which has a height of 318 m and a width of 44 m. The HFFB technique with four corner modifications is employed to carry out the aerodynamic optimization. The HFFB models and the surrounding buildings within a diameter range of 1000 m were fabricated at a geometric scale of 1:400. Wind tunnel tests were conducted in the ZD-1 boundary layer wind tunnel at Zhejiang University, China. The wind tunnel has a testing section 4.0 m wide and 3.0 m high. The geometrical scale ratio is carefully determined to ensure a small blockage ratio so as to diminish the distortion effects caused by the blockage in the testing section. It is found that the largest blockage ratio is less than 4.5%, indicating that the distortion effects on the flow in the testing section are negligible [21,22].

The model setup and wind direction are illustrated in Figure 1. The relative width r/B, defined as the roundness/recession radius r to the building width B, is taken as 5% and 10%, which is generally used in architectural design. The wind tunnel model configurations are illustrated in

Table 2. Corner Modifications.

	Case1	Case2	Case3	Case4	Case5
Section outline
Plan	Reference	5% roundness	10% roundness	5% recession	10% recession
Abbreviation	Ref	RO5%	RO10%	RE5%	RE10%
Model picture

, in which the five models with various corner modifications are denoted as Ref, RO5%, RO10%, RE5%, and RE10%.

The tests were conducted in the ZD-1 boundary layer wind tunnel at Zhejiang University, China. According to the Chinese code entitled “Load code for the design of building structures” [23], the terrain condition around the building is Terrain Category B, which corresponds to the open type terrain. In order to simulate the boundary layer wind field corresponding to Terrain Category B, spires arrays combined with floor roughness elements were employed in the wind tunnel (see Figure 1a). The wind velocities along the height at the center of the turntable were recorded using a three-dimensional Cobra Probe (Turbulent Flow Instrumentation). The mean wind speed v and turbulence intensity I profiles of Terrain Category B were tested in the wind tunnel and shown in Figure 2, in which the wind speed normalizes the mean wind speed at 350 m in full scale. It can be seen that the simulated profiles agree well with that regulated in GB 50009-2012 [23].

The HFFB model was mounted on a six-component force balance made by ME-β system company, Germany, with a measurement accuracy of 0.3% F.S. All models were made of lightweight wood, and the first sway frequency of the model-balance system is larger than 40 Hz, which meets the model requirement of the HFFB test. The base overturning moment and dynamic response can be attained after considering the dynamic properties of the building, such as mass and damping [24,25]. The wind tunnel tests were conducted at wind directions of 0–345° with an interval of 15°. The testing velocity atop the building was 10.06 m/s, and the HFFB data were recorded at a sampling frequency of 500 Hz with a sampling length of 60 s. The design wind velocity of the building was 30.98 m/s at the height of 10 m, and the wind forces in the following sections are presented in full scale based on the similarity criterion.

3. Aerodynamic Forces on the Building

3.1. Mean Base Overturning Moment

The mean base overturning moments M_x and M_y in the x-direction and y-direction of the models with various corner modifications are presented in Figure 3. It can be found that the absolute maximum M_x of the Ref model occurs at azimuth 0°. However, the absolute maximum M_x of the Ref model does not present a nearly equaled value at azimuth 180°. This is mainly due to the shielding effect induced by the neighboring buildings. Among the five models, the RO10% model shows the smallest value at azimuth 0° with a reduction rate of 32.7% compared with the Ref model, which is attributed to the fact that the roundness in the corner can reduce the positive pressure on the windward and the negative pressure on the leeward [2]. The most unfavorable angle of M_y lies at azimuth 270°. Similarly, the RO10% model reveals the smallest value among the five models with a reduction rate of 31.4% compared with the Ref model at this azimuth.

3.2. Standard Deviation of Fluctuating Base-Overturning Moment

The standard deviations of the base overturning moments M_x and M_y in the x-direction and y-direction of the five models are shown in Figure 4. It is found that the maximum standard deviation of M_x occurs at azimuth 270°. The pronounced standard deviation at the azimuth is mainly led by the cross-wind fluctuations originating from wake excitation. Among the five models, the RO10% model shows the smallest value at the azimuth. The standard deviations of M_x at azimuth 0° and 180° are much lower than that at azimuth 270°, which corresponds to the along wind fluctuating aerodynamic forces typically led by the approaching turbulence.

Furthermore, even in the along wind direction, the RO10% model shows the most beneficial effect in reducing the standard deviation of moments, indicating that the curved surfaces produce a less intensified vortex than the sharp edges. The modification of altering the edges can disrupt the vortex shedding process and mitigate the strength of cross-wind excitation forces [7]. The most unfavorable angle of the standard deviation of M_y lies at azimuth 0°, and the Ref model presents the maximum value among the five models. Similarly, the RE10% model shows the most favorable effectiveness in attenuating the fluctuating base overturning moment.

3.3. Spectrum of Fluctuating Base Overturning Moment

The standard deviations of the base overturning moments (see Figure 4) reveal that the critical azimuth for M_x and M_y are azimuth 270° and 0°, respectively, which corresponds to the cross-wind direction. Figure 5 shows the moment spectra in the critical azimuths, where f is the frequency and S_M is the base overturning moment spectrum. It can be observed that the spectra distribution of the five models varying with the azimuth and model are quite similar for the M_x and M_y. The results show that the critical reduced frequencies associated with the vortex shedding will increase when the corner modifications are made. It is worth noting that the peak values of the PSDs magnitude decrease significantly for the four models with corner modifications compared with the Ref model. Furthermore, the RO10% model among the four corner modifications shows the smallest values and appears to be the most effective in reducing the magnitude of vortex shedding.

4. Wind-Induced Response of the Building

4.1. Acceleration and Displacement

The equation of motion in the modal coordinate is expressed as:

$${q^{″}}_j(t)+2{\xi }_j{\omega }_j{q^{\prime }}_j(t)+{\omega }_j^2{q}_j(t)=\frac{P_j\left(t\right)}{M_j}$$

(1)

where ${q^{″}}_j$, ${q^{\prime }}_j$ and $q_j$ are the generalized acceleration, velocity and displacement of j-th mode, respectively; ${\xi }_j$ is the structural damping ratio of j-th mode and set as 0.04 [26]; ${\omega }_j$ is the circular natural frequency of j-th mode; and $P_j$ is the generalized aerodynamic forces of j-th mode.

$M_j$ is the generalized mass of j-th mode, which can be calculated as:

$$M_j={\displaystyle {\int }_0^{H}m(z){\mathrm{\Phi }}_{{}_j}^2(z)dz}$$

(2)

where $m$ is the structural mass along the building height; ${\mathrm{\Phi }}_j$ is the mode shape of j-th mode.

The variance of displacement and acceleration under wind excitation can then be obtained as:

$${\sigma }_{q_j}^2={\displaystyle \underset{0}{\overset{\infty }{\int }}{S}_{q_j}}\left(f\right)\mathrm{d}f=\frac{1}{{(2\pi f_j)}^4{M}_j{}^2}{\displaystyle {\int }_0^{\infty }{\left|H_j(f)\right|}^2{S}_{P_j}\left(f\right)}\mathrm{d}f$$

(3a)

$${\sigma }_{q_{{}_j}^{"}}^2={\displaystyle \underset{0}{\overset{\infty }{\int }}{(2\pi f)}^4{S}_{q_j}}\left(f\right)\mathrm{d}f=\frac{1}{{(2\pi f_j)}^4{M}_j{}^2}{\displaystyle {\int }_0^{\infty }{(2\pi f)}^4{\left|H_j(f)\right|}^2{S}_{P_j}\left(f\right)}\mathrm{d}f$$

(3b)

where $S_{p_j}$ is the spectrum of the j-th modal force spectrum and $f_j$ is the natural frequency of the j-th mode.

The wind-induced acceleration a and displacement d can then be determined on the basis of the dynamic structural properties of the building. According to the structural design, the first-order frequency f₁ equals 0.140 Hz in the x- and y- direction. The mass properties along the height are obtained from the design procedure.

Figure 6 shows the standard deviation of the acceleration a atop the building with various corner modifications. For the acceleration in the x-direction, azimuth 0° is the most unfavorable angle, corresponding to the cross-wind direction. At azimuth 0°, the standard deviation of acceleration decreases with the sequence of Ref, RO5%, RE5%, RE 10%, and RO10%. Obviously, the RO10% model shows the smallest values among the five models. For the acceleration in the y-direction, azimuth 270° is the most unfavorable angle, which corresponds to the cross-wind direction. The standard deviation of acceleration at azimuth 270° decreases with the sequence of Ref, RO5%, RE 10%, RE5%, and RO10%. As expected, the RO10% model shows the smallest values among the five models. Figure 7 illustrates the standard deviations of the displacement d atop the building. It can be found that the distribution of the displacement is quite similar to that of the acceleration with respect to the wind azimuth and models. It can also be seen that the RO10% model is the smallest among the five models for the displacement of the building.

4.2. Wind-Induced Base Overturning Moments

In addition to acceleration and displacement, the wind-induced responses of the structure also contain overturning moments. The base overturning moments consist of three parts: the mean load, the background component, and the inertia component induced by the building’s motions [2,11]. The correlation between the background component and the inertia component is weak. Therefore, the square root of the sum of the squares (SRSS) can be used for the combination [27]. The mean loads and the background component can be directly obtained from the tests, whereas the inertia part of the overturning moments can be obtained as:

$$M_I={\displaystyle {\int }_0^{H}m(z){\sigma }_{q″}\mathrm{\Phi }(z)zdz}$$

(4)

Figure 8 shows the extreme positive and negative base overturning moments M_x and M_y, in which the peak factor is taken as 2.5. It can be found that the cross-wind response dominates the results, i.e., M_x at azimuth 270° and M_y at azimuth 0°. At azimuth 270°, extreme positive and negative M_x decreases with the sequence of Ref, RO5%, RE5%, RE 10%, and RO 10%, which is consistent with displacement and acceleration. At azimuth 0°, extreme positive and negative M_y decreases with the sequence of Ref, RO5%, RE5%, RE 10%, and RO 10%. Once again, the RO10% model is the smallest among the five models.

Table 3. The comparison of wind-induced moments in y direction (Unit: 10⁹ N·m).

Case	Ref	RO 5%	RO 10%	RE 5%	RE 10%
Cross-wind (0°)	8.88	7.51	4.39	6.12	5.17
Along-wind (270°)	5.53	4.83	3.81	4.25	4.44

illustrates the comparison of wind-induced moments in y direction, the RO 10% shows the most beneficial effects in the wind-induced moments in both cross-wind and along-wind directions.

5. Comparisons with the SPI Test

5.1. Experimental Setup of SPI

The HFFB tests show that the RO10% model is the optimal scheme among the four corner modifications. Then, the SPI wind tunnel test on the RO10% model is conducted to verify the results of HFFB. Figure 9 shows the SPI testing model and its surroundings. There were 497 pressure taps installed on the surface of the model and tabulated in

Table 4. Layout of pressure taps.

Height	Tap Number	Height	Tap Number	Height	Tap Number
10 m	17	145 m	28	255 m	26
20 m	17	160 m	40	265 m	37
50 m	28	180 m	28	280 m	26
70 m	28	200 m	26	300 m	44
95 m	28	220 m	26	310 m	44
120 m	28	240 m	26

. The pressure taps are distributed on 17 levels along the height, and the taps on each lever were almost uniformly distributed around the cross-section. In addition, the taps are installed on the internal and external surfaces of the hung structures, such as the parapets and the tower crown, which suffer the wind forces on both surfaces. The DSM4000 (Scanivalve Corporation, Liberty Lake, WA, USA) was utilized to record the wind pressures on the model. The pressure taps were recorded at a sampling frequency of 312.5 Hz with a sampling length of 64 s.

5.2. Comparison of the Results of the HFFB and SPI Test

After the wind pressures on each tap are measured, the mean base overturning moments can be calculated by integrating the wind pressure on the model. Figure 10 shows the comparison between the mean M_x and M_y obtained by the SPI test and the HFFB test. It can be found that the results almost coincided with respect to the wind azimuth with acceptable discrepancy. The discrepancy may be attributed to several facts, such as an integrating error induced by the finite number of pressure and also a model fabrication difference between the HFFB model and the SPI model.

Figure 11 illustrates the extreme positive and negative base overturning moments M_x and M_y obtained using the SPI and HFFB tests, in which the calculating parameters such as dynamic properties of the building, damping ratio, and peak factor are the same. It can be found that both the positive and negative extreme moments are well matched with respect to the wind azimuth with acceptable discrepancy. The discrepancy can be partially attributed to facts such as the integrating error of SPI, model fabrications, and the calculating algorithm of equivalent static wind loading of the HFFB and SPI tests. Nevertheless, the discrepancy is slight, indicating that the two types of tests can be verified with each other, proving the reliability of the test results.

6. Conclusions

In this paper, a process with the combination of the HFFB technique to determine the most effective mitigation measure and the SPI technique to verify the effect was developed to provide an operable scheme for the aerodynamic optimization of an actual high-rise building. Then, the process is applied to a 318 m-tall high-rise building. The main conclusions of this study are as follows.

(1)

For the mean values of the base moments in the along-wind direction, the RO10% model reveals the smallest value among the five models. The results show that the mean base bending moment on the building will not be symmetrical about the wind azimuth due to the interfering effect induced by the neighboring buildings;

(2)

The standard deviation of the base bending moment of the building in the cross-wind direction is much larger than that in the along wind direction, indicating that the cross-wind fluctuations originating from vortex shedding will be dominated provided a remarkable building height;

(3)

The peak values of the power spectrum of the base bending moment decrease significantly for the models with corner modifications compared with the unmodified model. The aerodynamic treatments of corner modifications have a considerable benefit of disrupting the regular shedding of vortices and causing the cross-wind accelerations and displacements of buildings to be appreciably smaller than that of the unmodified model;

(4)

For the wind-induced base bending moments, the model with a 10% roundness radius to width ratio has the best mitigation effect not only in the along wind direction but also in the cross-wind direction among the four corner modifications;

(5)

Furthermore, the mean and extreme base overturning moments obtained by the SPI and the HFFB tests are almost coincidental versus the wind azimuth with acceptable discrepancy. The discrepancy may be partially due to facts such as the integrating error of the SPI technique and model fabrications. However, the disparities are relatively small, indicating that the two types of tests can be verified with each other.

The study shows that the 10% roundness radius to width ratio has the best mitigation wind effects of the actual building. It indicates if a reasonable corner modification of the high-rise building is applied, the wind load and wind-induced response in the along wind and cross-wind direction will be reduced significantly. As a result, remarkable economic savings will be expected. Therefore, the developed process in this paper is helpful and meaningful in the wind-resistant design for actual high-rise buildings.