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Vibration control of super-tall buildings using combination of tapering method and TMD system
Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
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Vibration control of super-tall buildings using combination of tapering method and TMD system Nahmat Khodaie Islamic Azad University, Khormouj Branch, P.O. Box: 75415/165, Khormouj, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Vibration control Wind-induced vibration Super-tall buildings TMD system Tapering effect
Modern super-tall buildings are highly sensitive to wind-induced vibrations due to their high flexibility and low inherent damping. Tapering the cross-section of buildings is an aerodynamic modification to control windinduced vibrations. Tuned mass damper (TMD) is another control system to mitigate structural vibrations. In the present study, the effect of using the tapering method together with the TMD system on the wind-induced vibration control of super-tall buildings is investigated. A numerical example of super-tall buildings is presented, and the wind-induced responses are computed for four different conditions, including: tapered and nontapered super-tall buildings with and without TMD system. The along-wind and crosswind responses are computed using the frequency domain analysis for different values of taper and TMD mass ratio. The results indicate that for tapered buildings, the natural frequency of the structure increases as taper ratio increases and since the acceleration is proportional to the square of the natural frequency, tapering method cannot effectively reduce the acceleration response. TMD has higher acceleration control performance than the displacement. By using the two control strategies together, both the displacement and acceleration responses can be effectively suppressed, and this method would be helpful in satisfying the occupant comfort and safety criteria of super-tall building.
1. Introduction In the last decades, the construction of extremely tall buildings has become possible due to the developments of lightweight and highstrength materials, progress in new structural systems and developments in analysis and design techniques. High level of flexibility and low inherent damping of modern super-tall buildings make them more susceptible to wind-induced vibrations. The excessive windinduced vibrations may cause discomfort or even fear to the occupants (Zhang and Li, 2018). Consequently, vibration control has attracted much interest from the application viewpoint to satisfy the serviceability and safety requirements. Kareem et al. (1999) classified the suppression means of wind-induced vibration of tall buildings into three main groups: (a) design modifications to structural systems such as increasing building mass, stiffness or natural frequency; (b) auxiliary damping devices as addition of materials with energy dissipative properties, increasing building damping ratio or adding an auxiliary mass system to increase the damping; and (c) aerodynamic design via improving aerodynamic properties to reduce wind force coefficient. A TMD system is a passive vibration control device, which is attached to a primary structure to limit its vibration amplitude under dynamic
forces (Asami et al., 2002). Many theoretical and experimental studies have been carried out to investigate the wind-induced vibration mitigation of the TMD system. Kawaguchi et al. (1992) and Tsukagoshi et al. (1993) presented time-domain analysis to evaluate the wind-induced oscillations of a tall building with the TMD. Kawaguchi et al. (1992) showed that for an example 42 m high building, the damper could reduce the vibration of primary mode around 60% when the mass ratio to the primary effective modal mass is 0.5%. Xu et al. (1992) investigated the effect of application of tuned liquid column and mass dampers in reducing the along-wind response of structures. A series of wind tunnel experiments and frequency-domain analytical approaches was carried out to study the performance of the TMD (Xu et al., 1992; Xu and Kwok, 1994). Ghorbani-Tanha et al. (2009) studied the effectiveness of TMD on the suppression of wind-induced motion of Milad Tower. The results showed that a 400-ton TMD system could reduce the RMS acceleration and displacement by 60% and 56% respectively. Tuan and Shang (2014) evaluated the performance of a 660-ton TMD in the Taipei 101 tower. The results indicated that the acceleration responses in the along-wind and crosswind directions are reduced by 31.7% and 33.8%, respectively. Recently, some researchers proposed a new structural system that has the self-control ability. In this system, a part of the main structure
E-mail address: [email protected]. https://doi.org/10.1016/j.jweia.2019.104031 Received 16 June 2019; Received in revised form 24 September 2019; Accepted 5 November 2019 Available online xxxx 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
performs as a large TMD system. Feng and Mita (1995) and Chai and Feng (1997) proposed the controlled Mega-Sub configuration. In the proposed system, the substructures, which each of them contain several floors, act as a massive TMD system. Some other researchers (Zhang et al. (2005), Zhang et al. (2009), Limazie et al. (2013) have investigated the vibration control performance of the Mega-Sub system. Wang et al. (2011) proposed the structural configuration of core-tubes with semi-flexible suspension systems to reduce seismic responses of structures. Liu and Lu, 2014 investigated the performance of suspended buildings under gravity and ground earthquake motion loading. Eimani and Khodaie, 2018 proposed a new structural system to reduce the wind-induced vibrations of super-tall buildings. The control performance of the proposed system was compared with that of the TMD system. This study showed that for the TMD mass ranged between 200 and 1000-ton, the average reduction of crosswind acceleration is about 66% more than that of the displacement. Amin and Ahuja (2010) classified the aerodynamic modifications in two groups: (a) minor modifications such as slotted or chamfered corners and roundness of corners, and (b) major modifications such as setbacks, tapering or twisting along the height. The tapering method alters the flow pattern around the building and reduces the crosswind vibration by inhibiting the formation of coherent wake fluctuations (Shimada, 1995; Karim, 1983). Kim and You (2002) and You et al. (2008) investigated the tapering effect on the reduction of crosswind responses of tall buildings through wind-tunnel tests on rigid and aero-elastic models. The model, which was corresponding to an actual building with height of 160 m and cross-section of 40 m 40 m, was produced to a scale of 1/400. Six types of taper ratios including zero, 2⋅5%, 5%, 7.5%, 10% and 15% were investigated. They concluded that the displacement and acceleration responses did not reduce significantly with a taper of 2.5–7⋅5%, but showed the greatest reduction with a taper of 15%. For the suburban terrain, the maximum reduction ratios of RMS displacement of along-wind and crosswind directions were 50% and 15% respectively. These values for RMS accelerations were 60% and 40% respectively. Authors also noted that the RMS acceleration might not be reduced even when the taper ratio is increased, which might be due to the fact that dynamic properties of the building, design wind speed and the variation of normalized power spectral density function also affect RMS acceleration. Kim and Kanda (2013) investigated the pressure fluctuations applied to the height-modified tall buildings through a series of wind pressure measurements. They found that taper and setback affect the bandwidth of power spectra of pressure coefficients and position of peak frequencies and through taper and setback, the height at which the vortex begins to form moves up. Deng et al. (2015) conducted a series of wind tunnel tests on tapered super high-rise buildings with a square cross-section. The height and the bottom width of the studied buildings were 546 m and 60 m respectively, and the top width varied between 24 and 60 m. They found that changing cross-section along with height can effectively reduce the crosswind aerodynamic loads for super high-rise buildings, and with increased taper ratio, the mitigation effect becomes greater. The effectiveness of aerodynamic modification to reduce wind loads has been widely reported in the previous studies (Hayashida and Iwasa (1990); Dutton et al. (1990); Amano (1995); Cooper et al. (1997); Kim and Kanda (2010); Kim et al. (2011); Tanaka et al., 2012). According to the literature, one of the important results of using TMD system is that this device has higher acceleration control performance than the displacement response. For the tapered tall building, because of lower effective modal mass and higher stiffness, the natural frequency of tapered buildings is higher than that of non-tapered buildings. Due to this result, since the acceleration response is proportional to the square of the natural frequency, tapering cannot effectively reduce the acceleration response. In the present study, the effect of combination of the two control strategies, i.e., tapered cross-section and TMD system is investigated on the wind-induced vibration control of super-tall buildings. A numerical example of super-tall building is presented, and the crosswind and along-wind responses are obtained for the structural and control
Fig. 1. General configuration of the studied tall buildings: (a) non-tapered; (b) non-tapered equipped with TMD; (c) tapered; and (d) tapered equipped with TMD system.
Fig. 2. General configuration of the tapered structure with TMD system, and its analytical model.
conditions shown in Fig. 1 for different values of taper and TMD mass ratio. It is demonstrated that by using the two control strategies, high level of response reduction can be achieved for both displacement and acceleration responses and this result would be helpful in satisfaction of serviceability conditions and occupant comfort criteria of super-tall buildings under wind loads. 2. Analytical models Fig. 1 shows the general configurations of the studied tall buildings, which includes the following conditions: (a) uniform cross-section, (b)
2
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
E x2i ¼
uniform cross-section and TMD system, (c) tapered cross-section and (d) tapered cross-section and TMD system. The structural behavior of tall building under wind loads is regarded as a vertical cantilever beam. This assumption is widely used for the analysis of wind-induced response of tall buildings (Samali et al. (2004) Wu and Yang, 1998 Ghorbani-Tanha et al. (2009); Eimani and Khodaie (2018); Khodaie, 2019). The stiffness matrix of the structure is obtained by assembling all stiffness matrices of the beam elements. The condensation method introduced by Paz, 1997 is carried out to condense out the rotation deformations to simplify the calculation procedure. The effect of the TMD system on the dynamic characteristics of the structure, and basic equations for obtaining the matrix of frequency response functions (FRFs) and structural responses are described in the next sections.
Z
þ∞ ∞
½0n1 mTMD
½Cs nn C¼ ½01n
½0nðn1Þ ½0n2 ½0n1 þ ; ½01ðn1Þ ½CTMD 22 0
z α
(8)
b
3.2. Mean and turbulent wind loads in the along-wind direction
KTMD ¼
(2)
Wind force in the along-wind direction is usually described as the sum of a mean part, assumed as constant within a conventional time interval, and a fluctuating part representing the atmospheric turbulence, which is usually modeled as a stationary zero-mean Gaussian random process. The mean part f D and the fluctuating part f ’D ðtÞ of the drag force per unit height can be stated as follow (Balendra, 1993):
1 2 f D ¼ ρa U CD B ; 2
þkTMD kTMD kTMD kTMD
þcTMD cTMD CTMD ¼ cTMD cTMD
f ’D ðtÞ ¼ ρa UuðtÞCD B
2
SfD ðωÞ ¼ ρ2a U B2 CD 2 Su ðωÞ
(4)
(10)
where Su ðωÞ is the spectral density of the turbulent velocity. Davenport (1961) presented the following formula for the PSD of the longitudinal wind velocity fluctuations for the frequency range of interest of dynamic response of buildings:
where ζs and ζa are structural and aerodynamic damping ratios respectively.
2.2. FRF matrix and the structural responses
Su ðωÞ ¼
The FRF matrix between the generalized structural displacements and the external forces can be obtained using the following equation (Newland, 1996):
600ω π u10
2
2u* 2
2 4=3 jωj ω 1 þ 600 π u10
(11)
pffiffiffi where u* ¼ κu10 denotes the friction velocity; u10 is the reference mean wind speed at 10 m above the ground, and κ is the surface drag coefficient. The correlation of wind velocity fluctuations between two different points is computed using the following formula (Vickery, 1970):
(5)
where ω is the angular frequency and j is the imaginary unit. The response power spectral density (PSD) matrix for an MDOF system is computed as follows (Newland, 1996): Sx ðωÞ ¼ HðωÞSf ðωÞH ðωÞ
(9)
where ρa is the air density, U and u(t) are the mean and the turbulent component of the wind velocity in the along-wind direction; CD ¼ CwþCl is the drag coefficient, Cw and Cl are pressure and suction coefficients; B is the tributary length normal to the wind direction. The power spectral density of the fluctuating force in the along-wind direction is:
where Ks and Cs are stiffness and damping matrices of the building, respectively, and kTMD and cTMD are stiffness and damping constants of the TMD system, respectively. The damping matrix of the structure Cs is obtained using the proportional damping method. The total damping ratio can be obtained as follows:
1 HðωÞ ¼ ω2 M þ jωC þ K
(7)
(1)
(3)
ζt ¼ ζs þ ζa
ω4 Sxii ðωÞdω
where α, a, and b are constants depending on the terrain type, z is the height of interest and Ub is the basic wind velocity. The basic or reference wind velocity in most codes and standards is based on the wind measurements at 10 m height in an open terrain associated with different mean recurrence intervals and averaging times (Zhou et al., 2002).
½0nðn1Þ ½0n2 ½Ks nn ½0n1 þ ; ½01ðn1Þ ½KTMD 22 ½01n 0
∞
There are two kinds of wind-velocity profile descriptions, i.e., the logarithmic and the power law. The general form of the power law is (NBCC, 2010):
where Ms is the mass matrix of the building, and mTMD is TMD mass. The global stiffness and damping matrices can be obtained by assembling the effects of the building and TMD system: K¼
þ∞
3.1. Wind velocity profile
Fig. 2 shows the general configuration of the tapered structure equipped with TMD system, and its analytical model. The global mass matrix of the structure can be written as: ½Ms nn ½01n
Z
3. Wind characteristics
2.1. The effect of the TMD system
E €x2i ¼
and
where Sxii ðωÞ is the ith element on the diagonal of the response PSD matrix.
UðzÞ ¼ aUb
M¼
Sxii ðωÞd ω
8 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > <jωj cZ 2 zj zk 2 þ cy 2 yj yk 2 = cohjk ðωÞ ¼ exp ðzk Þ > > : 2π ; 1 U z þU
(6)
2
where Sf ðωÞ is the PSD matrix of the excitation forces and * stands for the conjugate transpose. Finally, the displacement and acceleration mean square responses for the ith degree of freedom (DOF) can be obtained as follows:
(12)
j
The suggested values of cy and cz for engineering calculations are 16 and 10, respectively. The aerodynamic damping ratio in the along-wind direction is ob3
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
tained using the following formula proposed by Huilan et al. (2012):
0:5 1 ρa B U H pffiffiffiffiffiffiffi ζs 0:15 CD 0:002 4π ρs D f0 BD B
Table 1 Parameters of the tapered super-tall building with taper ratio.λ ¼ 5%
ζ a ¼ λm
(13)
where ρa =ρs is the ratio of air density to mass density of building, B and D are plan dimensions in the crosswind and along-wind directions, respectively, f0 is the building natural frequency, U is the mean wind velocity, ζs is the structural damping ratio, CD is the drag coefficient. λm is a modification coefficient equal to 1, 0.71, 1.17 and 1.26 for the taper ratios 0, 1%, 3% and 5%, respectively. 3.3. The power spectra of crosswind force In the present study, the following power spectra formula proposed by Liang et al. (2002) is used to compute the crosswind response of the building: nSðnÞ
σ
2
¼A
i 3 HðC1 Þ C 0:5 2 n þ ð1 AÞ 2 2 2 2 2 2 ð1 n Þ þ C1 n 1:56 ð1 n Þ þ C2 n
(14)
in which, σ ¼ 12ρa U 2 ðzÞCL B is the RMS of crosswind force at height z; CL is the mean RMS lift coefficient; B is the breath of windward side; n ¼ n= ns , ns ¼ St UðzÞ=B is the frequency of vortex shedding, St is the Strouhal number; A, C1, C2, and H(C1) are parameters function of the side ratio, height and cross-section area of building. The co-spectra of the crosswind forces between two different levels can be computed using the following coherence function: 2 Δ cohjk ¼ exp ;
α2
Δ¼
zj zk B
AL R5V C 2L R2V ρa B 2 ð1 RV Þ þ SL REVL ρs X
ðRV 1Þ
Height(m)
width(m)
Moment of inertia (m4)
Area (m2)
Mass (ton)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 336 352 368 384 400
49.2 48.4 47.6 46.8 46.0 45.2 44.4 43.6 42.8 42.0 41.2 40.4 39.6 38.8 38.0 37.2 36.4 35.6 34.8 34.0 33.2 32.4 31.6 30.8 30.0
8083 7494 6931 6392 5879 5390 4927 4489 4075 3687 3324 2986 2673 2385 2122 1884 1671 1483 1320 1183 1070 982 920 882 870
787.2 774.4 761.6 748.8 736.0 723.2 710.4 697.6 684.8 672.0 659.2 646.4 633.6 620.8 608.0 595.2 582.4 569.6 556.8 544.0 531.2 518.4 505.6 492.8 240.0
9683 9370 9063 8761 8464 8172 7885 7604 7327 7056 6790 6529 6273 6022 5776 5535 5300 5069 4844 4624 4409 4199 3994 3795 1800
building
height is assumed in the quadratic form: 2 z IðzÞ ¼ ðIbase Itop Þ 1 h þ Itop , where Ibase and Itop are the moment of inertia of the base and top-floor of the example building, respectively. The value of Itop is assumed about 10% of Ibase , and the value of Ibase is determined in such a way that the peak drift index of the building under the assumed wind loading is close to the allowable drift limit 1/600. The peak acceleration is generally taken to be less than 20 mili-g (20 cm/s2) to meet the occupant comfort requirement (Balendra, 1993). The allowable RMS acceleration can be obtained using the general expression pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of peak factor: gp ¼ 2 lnðνTÞ þ 0:5772= 2 lnðνTÞ, where T is the averaging time and ν is the up-crossing rate that is approximately equal to a natural frequency (Hz) of the building (Kwon and Kareem, 2013). For the studied buildings, the peak factor is about 3.75. Therefore, the allowable RMS acceleration is about 5.33 cm/s2. The results show that the obtained crosswind RMS acceleration for the uncontrolled structure is 14.14 cm/s2, which is very higher than the allowable limit and needs to be controlled. The taper ratio is defined as the difference of the top and bottom widths divided by the height of the building. For the tapered tall building, it is assumed that the average value of the top and base widths is equal to the width of the non-tapered building. By this assumption, the bottom width increases by increasing the taper ratio. This leads to increasing the stiffness and decreasing the lateral displacement of the taper buildings. It is assumed that the stiffness of the building is linearly proportional to the bottom width of the building. For instance, for the taper ratio 5%, which is equivalent to the base and top widths 50 and 30 m, respectively, the stiffness matrix of the building is obtained by multiplying that of the non-tapered building by the bottom width ratio 1.25. By this assumption, the results indicate that the top-floor reduced wind velocity, which has an important role in the crosswind response, are equal to each other for the studied buildings. Table 1 presents the structural properties of the typical tapered building with the taper ratio λ ¼ 5% for a 25 D.O.F analytical model, including height, width, moment of inertia, lumped mass, and tributary area of each node. The critical inherent damping ratio for the first mode is assumed ζs1 ¼ 0.01, and the damping ratios for the higher modes are calculated using the following equation (Kareem, 1996):
(15)
in which α2 is a constant, which is equal to 5.56 for the side ratio D/B between 0.25 and 1. The aerodynamic damping ratio in the crosswind direction is obtained using the formula proposed by Gu et al. (2014): ζa ¼
Node
(16)
where RV ¼ Vm =Vs ; Vm is the reduced wind velocity of building [Vm ¼ UH =ðf0 BÞ; Vs is the reduced wind velocity of Strouhal frequency [Vs ¼ UH =ðfS BÞ; B/X is the ratio of building width to the wind-induced displacement response, AL , CL , SL and EL are parameters function of the side ratio, turbulent intensity, aspect ratio and taper ratio. For instance, typical values of AL , CL , SL ; and EL for the building with uniform square cross-section, with aspect ratio 8 and turbulent intensity 11%, are 0.037, 0.89 and 0.202 and 4.62, respectively. 4. Numerical study A numerical study is presented to demonstrate the control effectiveness of combination of tapering method and TMD system. The structures are modeled as a multi-degrees-of-freedom vertical cantilever beam. The PSD matrices of the along-wind and crosswind loads are obtained using the equations presented in Sect. 3, and the frequency-domain analysis, described in Sect. 2, is used to compute the along-wind and crosswind RMS displacement and acceleration values. The effect of different values of taper ratios and TMD mass on the control performance of the structure is investigated. The physical properties of the example super-tall building and the assumed aerodynamic information are presented in the following paragraphs. The super-tall building is assumed with the height H ¼ 400m and square cross-section. Width and depth for non-tapered building were considered 40m. The bulk mass density of the building is assumed ρs ¼ 250 kg/m3. The variation of the moment of inertia along the 4
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
Fig. 3. Variation of the top-floor along-wind and crosswind responses versus basic wind speed: (a) RMS displacement, (b) RMS acceleration.
fi ζ si ¼ ζs1 1 þ 0:38 1 f1
(17)
Table 2 Optimum TMD parameters and the corresponding along-wind and crosswind responses.
where f1 and fi are the first ant ith natural frequency of the structure. The aerodynamic physical data are chosen as follows: air density ρa ¼ 1.25 kg/m3; 1-h average basic wind speed Ub ¼ 30 m/s, which is equivalent to 3-sec gust wind speed of 47.4 m/s; gradient height zg ¼ 383 m; exponent for mean wind power α ¼ 0.36. The drag coefficients is assumed CD ¼ 1.3 as proposed by Canadian code (NBCC, 2010) for tall buildings with square cross-section. The surface drag coefficient is assumed κ ¼ 0.03. The crosswind PSD matrix parameters appeared in Sect. 3.3 are obtained as follows: A ¼ 0.96, C1 ¼ 0.019, C2 ¼ 2.0, H(C1) ¼ 0.093, α2 ¼ 5:56, the mean RMS lift coefficient CL ¼ 0.404, Strouhal number St ¼ 0.084 (Liang et al., 2002).
TMD mass Ratio μn
0.0% 0.4% 0.8% 1.2% 1.6% 2.0%
Using the procedure described in Sections 3 and 4, the along-wind and crosswind responses are obtained for the studied structures presented in Fig. 1. The assumptions and the results for each structural condition are presented in the following sections. Since the greatest wind-induced responses occur at the top-floor of the structure, the topfloor RMS displacement and acceleration values are presented and discussed herein. In order to evaluate and compare the control performance of the structures, the control effectiveness is defined as the percentage reduction of the top-floor RMS response compared to that response of the non-tapered building, which can be stated as follows:
fdopt
σ xa (cm)
0.0385 0.0541 0.0659 0.0756 0.0841
0.996 0.992 0.988 0.984 0.980
σ x€a (cm/
σ xc (cm)
s2) 7.74 6.34 6.01 5.82 5.69 5.59
7.22 5.37 4.89 4.60 4.40 4.24
σ x€c (cm/ s2)
15.54 13.15 12.55 12.21 11.99 11.83
14.14 10.84 9.91 9.35 8.95 8.65
In this section, wind-induced responses are presented for the example super-tall building equipped with TMD systems having different values of mass. The frequency ratio fd and the damping ratio ζd for the TMD are defined as:
(18)
where σ u and σ c are the top-floor RMS responses of the uncontrolled and controlled structures, respectively.
fd ¼ ωd =ωn
5.1. Non-tapered super-tall building (uncontrolled structure)
and
ζd ¼ cd =ð2md ωn Þ
(20)
where md , cd , and ωd are the mass, damping, and frequency of the TMD system, and ωn is the lowest natural frequency of the building. The TMD tuning formulas presented by Hahnkamm, 1932 and Brock, 1946 are used to optimize the response of the building:
Fig. 3(a) and (b) show the variation of the top-floor along-wind and crosswind RMS displacement and acceleration responses, respectively, versus basic wind speed. These figures show that the crosswind responses are higher than the along-wind responses. For the presented values of the basic wind speed, difference between the crosswind and along-wind responses increases as the wind speed increases. The crosswind force is due predominantly to the fluctuating lift force induced by the vortices shed in the building’s wake (Simiu, 2011). The resonant or critical wind velocity, at which the vortex shedding frequency coincides with the natural frequency of the building, can be obtained using the following equation: Ucr ref ¼ Bref f 0 St
ζd opt
Crosswind
5.2. Non-tapered super-tall building with TMD system
σu σc *100 σu
Along-wind
equal to 0.084 for the square cross-section (Liang et al., 2002), and f 0 is the dominant frequency of the structure equal to 0.166 for the non-tapered super-tall building. According to this equation, the critical top-floor wind speed for the studied non-tapered building is about 79 m/s. For the basic wind speed Ub ¼ 30 m/s, the top-floor wind velocity from the power law is UH ¼ 47.4 m/s, which is lower than the resonant wind velocity as expected. For this condition, the top-floor responses are as follows: the static displacement 22.22 cm, the along-wind and crosswind RMS displacements σ xa ¼ 7.74 and σ xc ¼ 15.54 cm, respectively, and the along-wind and crosswind RMS accelerations σ x€a ¼ 7.22 and σ x€c ¼ 14.14 cm/s2, respectively. This result shows that for the example
5. Analysis and results
Control effectiveness ¼
Optimum parameters
fd opt ¼
1 1þμ
and
ζ dopt ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3μ 8ð1 þ μÞ3
(21)
where μ is the mass ratio of TMD to the generalized mass of first mode of the building. Table 2 presents the optimum TMD parameters and the corresponding optimized along-wind and crosswind RMS displacements σ xa and σ xc , and RMS accelerations σ x€a and σ x€c , respectively. The TMD mass ratio μn is defined as the mass ratio of TMD to the primary modal mass of the nontapered tall building. The maximum value of μn is assumed 2%, which is
(19)
where Bref is the reference crosswind width, St is the Strouhal number 5
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
The first natural frequency of the tapered buildings are presented in Table 3 for different values of λ. As it is expected, the natural frequency of the structure increases as the taper ratio increases due to decreasing the effective modal mass and increasing the stiffness of the tapered buildings. The acceleration response is directly proportional to the displacement response and square of the natural frequency. Therefore, the acceleration response does not show considerable change versus variation of taper ratio due to increasing the natural frequency. Both along-wind and crosswind acceleration responses increase for taper ratio λ ¼ 1% compared to the corresponding responses of non-tapered building. The along-wind acceleration in the range λ ¼ 1% to 5% decreases by increasing the taper ratio. However, the reduction percentage is considerably lower than that of the displacement response. For instance, the along-wind acceleration reduction for λ ¼ 5% is 5.4%. The crosswind accelerations of tapered building due to lower aerodynamic damping are higher than that of non-tapered building. Fig. 5 shows the variation of the crosswind acceleration and displacement reduction percentages versus the taper ratio. This figure shows that the aerodynamic modification of tapering along the building height can effectively reduce displacement of the structure. However, the acceleration response of tapered building show 0–5.5% increase compared to that response of non-tapered building, depending on the taper ratio. The maximum increase is related to the taper ratio λ ¼ 1%. Therefore, tapering method is not effective in controlling the acceleration response.
Fig. 4. Variation of the control effectiveness of the crosswind acceleration and displacement versus μn .
equivalent to a 686-ton TMD system. The results show that the control effectiveness of the TMD system increases as the TMD mass increases for both along-wind and crosswind responses. For μn ¼ 2%, the along-wind and crosswind RMS displacements decrease by 27.8% and 23.9%, respectively, compared to the corresponding responses of the uncontrolled structure. The along-wind and crosswind response reduction values for the RMS accelerations are 41.3% and 38.8%, respectively. This result indicates that the control effectiveness of the TMD system for the along-wind and crosswind responses are approximately close to each other. Fig. 4 shows the variation of the crosswind acceleration and displacement reduction percentages versus TMD mass ratio μ. This figure indicates that the control effectiveness of the TMD system for the acceleration is higher than that for the displacement. For example, the crosswind acceleration and displacement reduction percentages for μn ¼ 2% are 38.8 and 23.9%, respectively. This figure also shows that the control effectiveness of the TMD system increases with increasing μn , while the slope of the variation curve decreases. For instance, the absolute value of the slope of the acceleration curve at μn ¼ 0:3% is about 7 times of that at μn ¼ 2%. This means that the higher control efficiencies of the TMD system require using very heavier mass dampers.
5.4. Tapered super-tall building with TMD system The control effectiveness of the combination of tapering method and TMD system for different values of TMD mass ratio and taper ratio are presented herein. Fig. 6 shows the variation of the crosswind displacement and acceleration reduction percentages for the tapered super-tall building with a TMD system with mass ratio μn ¼ 2% versus taper ratio. This figure indicates that using the control techniques of tapering
5.3. Tapered super-tall building without TMD system Wind-induced responses for the tapered super-tall building for different values of taper ratio λ between zero and 5% are presented and discussed in this section. Taper ratio is defined as the difference between the top and bottom widths divided by the height of the building. Table 3 presents the top-floor along-wind and crosswind RMS responses, as well as, the first natural frequency of the super-tall buildings and the alongwind and crosswind aerodynamic damping ratios for different values of λ. The results show that the along-wind and crosswind displacement responses decrease with increasing λ. For λ ¼ 5%, the along-wind and crosswind RMS displacement decrease by 42.6% and 41.7% compared to the corresponding responses of the non-tapered building, respectively.
Fig. 5. Variation of the control effectiveness of the crosswind acceleration and displacement versus.λ
Table 3 Top-floor along-wind and crosswind responses versus taper ratio for the tapered buildings. Taper ratio λ
0 1% 2% 3% 4% 5%
Natural frequency
0.166 0.176 0.186 0.197 0.209 0.221
Along-wind
Crosswind
ζ a ð%Þ
σ xa (cm)
σ x€a (cm/s )
ζ a ð%Þ
σ xc (cm)
σ x€c (cm/s2)
0.39 0.27 0.36 0.44 0.46 0.49
7.74 7.13 6.26 5.52 4.95 4.44
7.22 7.49 7.21 6.95 6.88 6.80
0.55 0.32 0.33 0.31 0.27 0.19
15.54 14.41 12.67 11.24 10.03 9.06
14.14 14.92 14.59 14.43 14.39 14.59
2
6
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
structure increases with increasing μn and λ. The maximum control effectiveness, which can be achieved for μn ¼ 2% and λ ¼ 5%, are 58.24 and 46.18% for the displacement and acceleration responses, respectively. Table 4 presents the crosswind RMS displacement and acceleration values and the corresponding reduction percentages for different structural condition including non-tapered building, tapered building (λ ¼ 5%), non-tapered building with TMD (μn ¼ 2%), and tapered building equipped with TMD (λ ¼ 5%, μn ¼ 2% ). This table also shows the average reduction percentages of displacement and acceleration responses for each case. As we can see, the average response reduction for the tapered building without TMD is 19.26%, which increases to 52.21% when the TMD system is used. This result indicates that using the control strategies of tapering and TMD system together reduces more efficiently both acceleration and displacement responses compared to the conditions that only one of the control methods is used. This method can be very helpful in satisfying the occupant comfort criteria and safety requirements in design of super-tall buildings under wind loads.
Fig. 6. Variation of the reduction percentages of the crosswind acceleration and displacement for a tapered building with a TMD system for μn ¼ 2% versus taper ratio.λ
6. Conclusions
and TMD together can be effective in reducing both displacement and acceleration responses. The value μn ¼ 2% is equivalent to a 686-ton TMD system. By increasing the taper ratio, the primary modal mass of the structure decreases due to decreasing the volume of tapered building by height. Therefore, for a constant value of TMD mass, the effective mass ratio of the TMD increases that results in increasing the control effectiveness of the TMD system for taper buildings. The displacement and acceleration reduction curves intersect at λ ¼ 2.25%, which results in about 42% displacement and acceleration reduction. For λ ¼ 5%, the displacement and acceleration reduction values are 58.24% and 46.18%, respectively. Figs. 7 and 8 show the contour plot of reduction percentages of crosswind displacement and acceleration, respectively, versus different values of TMD mass ratio μn and taper ratio λ. The maximum values of μn and λ are assumed 2% and 5%, respectively. For every pair of μn and λ, the corresponding response reduction percentage can be obtained using the contour plots. For instance, for μn ¼ 1% and λ ¼ 3%, the reduction percentages of crosswind displacement and acceleration are 45 and 36%, respectively. These figures show that the control effectiveness of the
Tapering of building along the height has been studied as an aerodynamic modification method to reduce wind-induced vibrations. As reported in the previous studies, although this control method can effectively reduce the wind-induced drift of tall buildings, it is not efficient in reduction of the acceleration response. In the present study, tapering method and TMD system are used together to achieve higher control performance for the acceleration and displacement responses. In order to investigate the control performance of the studied systems, a numerical example of a 400-m high super-tall building is presented, and the responses are computed for the following conditions: (a) non-tapered building; (b) non-tapered building with TMD system; (c) tapered building and (d) tapered building with TMD system. The TMD mass ratio μn is defined as the mass ratio of TMD to the primary modal mass of the nontapered tall building. The maximum taper ratio and TMD mass ratio are assumed λ ¼ 5% and μn ¼ 2%, respectively, and the crosswind and alongwind responses are computed for different values of λ and μn using the frequency domain analysis of MDOF structures. For the controlled structures, the control effectiveness is defined as the percentage reduction in the building top-floor RMS response compared to that of the un-
Fig. 7. Contour plot of reduction percentages of the crosswind displacement versus different values of μn and λ. 7
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
Fig. 8. Contour plot of reduction percentages of the crosswind acceleration versus different values of μn and λ.
Table 4 The crosswind RMS displacement and acceleration values and the corresponding reduction percentages for different structural condition. Structural condition
Non-tapered Tapered (λ ¼ 5%) Non-tapered with TMD (μn ¼ 2%) Tapered with TMD (λ ¼ 5%, μn ¼ 2% )
Crosswind
Control effectiveness (%)
σ xc (cm)
σ x€c (cm/s )
Displacement
Acceleration
15.54 9.06 11.83 6.49
14.14 14.59 8.65 7.61
0 41.70 23.87 58.24
0 3.18 38.83 46.18
2
Average reduction (%)
0 19.26 31.35 52.21
Funding
controlled structure. The results indicate that for the studied uncontrolled super-tall building with the assumed structural and wind characteristics, the crosswind response is higher than the occupant comfort level and needs to be controlled. The first natural frequency of the tapered super-tall buildings increases with increasing λ, due to decreasing the effective modal mass and increasing the stiffness of the structure. For instance, the natural frequency of the building with λ ¼ 5% is 33% more than that of the non-tapered building. Since the acceleration response is proportional to the displacement response and square of the natural frequency, although tapering effect is efficient in reduction of displacement, it cannot effectively reduce the acceleration response due to increase of the natural frequency. For instance, the displacement reduction percent for the taper ratio λ ¼ 5% is 41.7%, while the acceleration response increased by 3.18%. For the non-tapered super-tall building equipped with TMD system, the control effectiveness of acceleration is higher than that of the displacement, and the vibration control effect increases as the TMD mass ratio increases. The displacement and acceleration control effectiveness for the non-tapered building with TMD mass ratio μn ¼ 2% are 23.87 and 38.83%, respectively. The highest control performance is observed for the tapered building with TMD system. For instance, for the structure with μn ¼ 2% and λ ¼ 5%, the displacement and acceleration control effectiveness are 58.24 and 46.18%, respectively. The average displacement and acceleration control effectiveness for the conditions including tapered λ ¼ 5%, non-tapered equipped with TMD μn ¼ 2%, and tapered equipped with TMD λ ¼ 5% and μn ¼ 2% are 19.26, 31.35 and 52.21%, respectively. These results demonstrate the efficient performance of using tapering method and TMD system together in reducing the wind-induced vibration of super-tall buildings, which can satisfy the safety and occupant comfort criteria under wind loads.
Funding was received for this work. All of the sources of funding for the work described in this publication are acknowledged below: Financial support from Iran National Science Foundation (INSF) through Grant. 97011187. Declaration of competing interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. Acknowledgments The author is grateful for the financial support from Iran National Science Foundation (INSF) through Grant. 97011187. References Amano, T., 1995. The effect of corner-cutting of three dimensional square cylinders on vortex-induced oscillation and galloping in uniform flow. J. Struct. Constr. Eng., AIJ 60 (478), 63–69. Amin, J.A., Ahuja, A.K., 2010. Aerodynamic modifications to the shape of the buildings: a review of the state-of-the-art. Asian J. Civ. Eng. 11 (4), 433–450. Asami, T., Nishihara, O., Amr, M., 2002. Analytical solutions to H∞ and H2 optimization of dynamic vibration absorbers attached to damped linear systems. J. Vib. Acoust. 124 (2), 284–295. Balendra, T., 1993. Vibration of Building to Wind and Earthquake Loads. Springer-Verlag, London. Brock, J.E., 1946. A note on the damped vibration absorber. ASME J. Appl. Mech. 13 (4), A-284. 8
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Chai, W., Feng, M.Q., 1997. Vibration control of super tall buildings subjected to wind loads. Int. J. Non-Linear Mech. 32 (4), 657–668. Cooper, K.R., Nakayama, M., Sasaki, Y., Fediw, A.A., Resende-Ide, S., Zan, S.J., 1997. Unsteady aerodynamic force measurements on a super-tall building with a tapered cross section. J. Wind Eng. Ind. Aerodyn. 72, 199–212. Davenport, A.G., 1961. The application of statistical concepts to the wind loading of structures. Proc. Inst. Civ. Eng. 19, 449–472. Deng, T., Yu, X., Xie, Z., 2015. Aerodynamic measurements of across-wind loads and responses of tapered super high-rise buildings. Wind Struct. 21 (3), 331–352. Dutton, R., Isyumou, N., 1990. Reduction of tall building motion by aerodynamic treatments. J. Wind Eng. Ind. Aerodyn. 36, 739–747. Eimani, H., Khodaie, N., 2018. Wind-induced vibration control of super-tall buildings using a new combined structural system. J. Wind Eng. Ind. Aerodyn. 172, 256–266. Feng, M.Q., Mita, A., 1995. Vibration control of tall buildings using mega sub configuration. J. Eng. Mech. 121 (10), 1082–1088. Ghorbani-Tanha, A., Noorzad, K., Rahimian, M., 2009. Mitigation of wind-induced motion of Milad tower by tuned mass damper. Struct. Des. Tall Special Build. 18, 371–385. Gu, M., Cao, H.L., Quan, Y., 2014. Experimental study of across-wind aerodynamic damping of super high-rise buildings with aerodynamically modified square crosssections. Struct. Des. Tall Special Build. 23, 1225–1245. Hahnkamm, E., 1932. Die dampfung von fundaments-chwingungen bei veranderlicher erregergrequenz. Ing Arch 4, 192–201. Hayashida, H., Iwasa, Y., 1990. Aerodynamic shape effects on tall building for vortex induced vibration. J. Wind Eng. Ind. Aerodyn. 33 (1–2), 237–242. Huilan, C., Yong, Q., Ming, G., Di, W., 2012. Along-wind aerodynamic damping of highrise buildings. In: The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China. Karim, A., 1983. Mitigation of Wind induced motion of tall building. J. Wind Eng. Ind. Aerodyn. 11, 273–284. Kareem, A., 1996. Damping in structures: its evaluation and treatment of uncertainty. J. Wind Eng. Ind. Aerodyn. 59, 131–157. Kareem, A., Kijewski, T., Tamura, Y., 1999. Mitigation of motions of tall buildings with specific examples of recent applications. Wind Struct. 3 (4), 201–251. Kawaguchi, A., Teramura, A., Omote, Y., 1992. Time history response of a tall building with a tuned mass damper under wind force. J. Wind Eng. Ind. Aerodyn. 41.44, 1949–1960. Kim, Y., Kanda, J., 2013. Wind pressures on tapered and set-back tall buildings. J. Fluids Struct. 39, 306–321. Khodaie, N., 2019. Evaluation of modal contribution to the wind‑induced responses of tall buildings. Asian J. Civ. Eng. 20 (6), 837–846. Kim, Y.C., Kanda, J., 2010. Effects of taper and setback on wind force and wind-induced response of tall buildings. Wind Struct. 13 (6), 499–517. Kim, Y.C., Kanda, J., Tamura, Y., 2011. Wind-induced coupled motion of tall buildings with varying square plan with height. J. Wind Eng. Ind. Aerodyn. 99 (5), 638–650. Kim, Y.M., You, K.P., 2002. Dynamic response of a tapered tall building to wind loads. J. Wind Eng. Ind. Aerodyn. 90 (12–15), 1771–1782. Kwon, D.K., Kareem, A., 2013. Comparative study of major international wind codes and standards for wind effects on tall buildings. Eng. Struct. 51, 23–35. Liang, S., Liu, S., Li, Q.S., Zhang, L., Gu, M., 2002. Mathematical model of acrosswind dynamic loads on rectangular tall buildings. J. Wind Eng. Ind. Aerodyn. 90, 1757–1770.
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Vibration control of super-tall buildings using combination of tapering method and TMD system Nahmat Khodaie Islamic Azad University, Khormouj Branch, P.O. Box: 75415/165, Khormouj, Iran
A R T I C L E I N F O
A B S T R A C T
Keywords: Vibration control Wind-induced vibration Super-tall buildings TMD system Tapering effect
Modern super-tall buildings are highly sensitive to wind-induced vibrations due to their high flexibility and low inherent damping. Tapering the cross-section of buildings is an aerodynamic modification to control windinduced vibrations. Tuned mass damper (TMD) is another control system to mitigate structural vibrations. In the present study, the effect of using the tapering method together with the TMD system on the wind-induced vibration control of super-tall buildings is investigated. A numerical example of super-tall buildings is presented, and the wind-induced responses are computed for four different conditions, including: tapered and nontapered super-tall buildings with and without TMD system. The along-wind and crosswind responses are computed using the frequency domain analysis for different values of taper and TMD mass ratio. The results indicate that for tapered buildings, the natural frequency of the structure increases as taper ratio increases and since the acceleration is proportional to the square of the natural frequency, tapering method cannot effectively reduce the acceleration response. TMD has higher acceleration control performance than the displacement. By using the two control strategies together, both the displacement and acceleration responses can be effectively suppressed, and this method would be helpful in satisfying the occupant comfort and safety criteria of super-tall building.
1. Introduction In the last decades, the construction of extremely tall buildings has become possible due to the developments of lightweight and highstrength materials, progress in new structural systems and developments in analysis and design techniques. High level of flexibility and low inherent damping of modern super-tall buildings make them more susceptible to wind-induced vibrations. The excessive windinduced vibrations may cause discomfort or even fear to the occupants (Zhang and Li, 2018). Consequently, vibration control has attracted much interest from the application viewpoint to satisfy the serviceability and safety requirements. Kareem et al. (1999) classified the suppression means of wind-induced vibration of tall buildings into three main groups: (a) design modifications to structural systems such as increasing building mass, stiffness or natural frequency; (b) auxiliary damping devices as addition of materials with energy dissipative properties, increasing building damping ratio or adding an auxiliary mass system to increase the damping; and (c) aerodynamic design via improving aerodynamic properties to reduce wind force coefficient. A TMD system is a passive vibration control device, which is attached to a primary structure to limit its vibration amplitude under dynamic
forces (Asami et al., 2002). Many theoretical and experimental studies have been carried out to investigate the wind-induced vibration mitigation of the TMD system. Kawaguchi et al. (1992) and Tsukagoshi et al. (1993) presented time-domain analysis to evaluate the wind-induced oscillations of a tall building with the TMD. Kawaguchi et al. (1992) showed that for an example 42 m high building, the damper could reduce the vibration of primary mode around 60% when the mass ratio to the primary effective modal mass is 0.5%. Xu et al. (1992) investigated the effect of application of tuned liquid column and mass dampers in reducing the along-wind response of structures. A series of wind tunnel experiments and frequency-domain analytical approaches was carried out to study the performance of the TMD (Xu et al., 1992; Xu and Kwok, 1994). Ghorbani-Tanha et al. (2009) studied the effectiveness of TMD on the suppression of wind-induced motion of Milad Tower. The results showed that a 400-ton TMD system could reduce the RMS acceleration and displacement by 60% and 56% respectively. Tuan and Shang (2014) evaluated the performance of a 660-ton TMD in the Taipei 101 tower. The results indicated that the acceleration responses in the along-wind and crosswind directions are reduced by 31.7% and 33.8%, respectively. Recently, some researchers proposed a new structural system that has the self-control ability. In this system, a part of the main structure
E-mail address: [email protected]. https://doi.org/10.1016/j.jweia.2019.104031 Received 16 June 2019; Received in revised form 24 September 2019; Accepted 5 November 2019 Available online xxxx 0167-6105/© 2019 Elsevier Ltd. All rights reserved.
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
performs as a large TMD system. Feng and Mita (1995) and Chai and Feng (1997) proposed the controlled Mega-Sub configuration. In the proposed system, the substructures, which each of them contain several floors, act as a massive TMD system. Some other researchers (Zhang et al. (2005), Zhang et al. (2009), Limazie et al. (2013) have investigated the vibration control performance of the Mega-Sub system. Wang et al. (2011) proposed the structural configuration of core-tubes with semi-flexible suspension systems to reduce seismic responses of structures. Liu and Lu, 2014 investigated the performance of suspended buildings under gravity and ground earthquake motion loading. Eimani and Khodaie, 2018 proposed a new structural system to reduce the wind-induced vibrations of super-tall buildings. The control performance of the proposed system was compared with that of the TMD system. This study showed that for the TMD mass ranged between 200 and 1000-ton, the average reduction of crosswind acceleration is about 66% more than that of the displacement. Amin and Ahuja (2010) classified the aerodynamic modifications in two groups: (a) minor modifications such as slotted or chamfered corners and roundness of corners, and (b) major modifications such as setbacks, tapering or twisting along the height. The tapering method alters the flow pattern around the building and reduces the crosswind vibration by inhibiting the formation of coherent wake fluctuations (Shimada, 1995; Karim, 1983). Kim and You (2002) and You et al. (2008) investigated the tapering effect on the reduction of crosswind responses of tall buildings through wind-tunnel tests on rigid and aero-elastic models. The model, which was corresponding to an actual building with height of 160 m and cross-section of 40 m 40 m, was produced to a scale of 1/400. Six types of taper ratios including zero, 2⋅5%, 5%, 7.5%, 10% and 15% were investigated. They concluded that the displacement and acceleration responses did not reduce significantly with a taper of 2.5–7⋅5%, but showed the greatest reduction with a taper of 15%. For the suburban terrain, the maximum reduction ratios of RMS displacement of along-wind and crosswind directions were 50% and 15% respectively. These values for RMS accelerations were 60% and 40% respectively. Authors also noted that the RMS acceleration might not be reduced even when the taper ratio is increased, which might be due to the fact that dynamic properties of the building, design wind speed and the variation of normalized power spectral density function also affect RMS acceleration. Kim and Kanda (2013) investigated the pressure fluctuations applied to the height-modified tall buildings through a series of wind pressure measurements. They found that taper and setback affect the bandwidth of power spectra of pressure coefficients and position of peak frequencies and through taper and setback, the height at which the vortex begins to form moves up. Deng et al. (2015) conducted a series of wind tunnel tests on tapered super high-rise buildings with a square cross-section. The height and the bottom width of the studied buildings were 546 m and 60 m respectively, and the top width varied between 24 and 60 m. They found that changing cross-section along with height can effectively reduce the crosswind aerodynamic loads for super high-rise buildings, and with increased taper ratio, the mitigation effect becomes greater. The effectiveness of aerodynamic modification to reduce wind loads has been widely reported in the previous studies (Hayashida and Iwasa (1990); Dutton et al. (1990); Amano (1995); Cooper et al. (1997); Kim and Kanda (2010); Kim et al. (2011); Tanaka et al., 2012). According to the literature, one of the important results of using TMD system is that this device has higher acceleration control performance than the displacement response. For the tapered tall building, because of lower effective modal mass and higher stiffness, the natural frequency of tapered buildings is higher than that of non-tapered buildings. Due to this result, since the acceleration response is proportional to the square of the natural frequency, tapering cannot effectively reduce the acceleration response. In the present study, the effect of combination of the two control strategies, i.e., tapered cross-section and TMD system is investigated on the wind-induced vibration control of super-tall buildings. A numerical example of super-tall building is presented, and the crosswind and along-wind responses are obtained for the structural and control
Fig. 1. General configuration of the studied tall buildings: (a) non-tapered; (b) non-tapered equipped with TMD; (c) tapered; and (d) tapered equipped with TMD system.
Fig. 2. General configuration of the tapered structure with TMD system, and its analytical model.
conditions shown in Fig. 1 for different values of taper and TMD mass ratio. It is demonstrated that by using the two control strategies, high level of response reduction can be achieved for both displacement and acceleration responses and this result would be helpful in satisfaction of serviceability conditions and occupant comfort criteria of super-tall buildings under wind loads. 2. Analytical models Fig. 1 shows the general configurations of the studied tall buildings, which includes the following conditions: (a) uniform cross-section, (b)
2
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
E x2i ¼
uniform cross-section and TMD system, (c) tapered cross-section and (d) tapered cross-section and TMD system. The structural behavior of tall building under wind loads is regarded as a vertical cantilever beam. This assumption is widely used for the analysis of wind-induced response of tall buildings (Samali et al. (2004) Wu and Yang, 1998 Ghorbani-Tanha et al. (2009); Eimani and Khodaie (2018); Khodaie, 2019). The stiffness matrix of the structure is obtained by assembling all stiffness matrices of the beam elements. The condensation method introduced by Paz, 1997 is carried out to condense out the rotation deformations to simplify the calculation procedure. The effect of the TMD system on the dynamic characteristics of the structure, and basic equations for obtaining the matrix of frequency response functions (FRFs) and structural responses are described in the next sections.
Z
þ∞ ∞
½0n1 mTMD
½Cs nn C¼ ½01n
½0nðn1Þ ½0n2 ½0n1 þ ; ½01ðn1Þ ½CTMD 22 0
z α
(8)
b
3.2. Mean and turbulent wind loads in the along-wind direction
KTMD ¼
(2)
Wind force in the along-wind direction is usually described as the sum of a mean part, assumed as constant within a conventional time interval, and a fluctuating part representing the atmospheric turbulence, which is usually modeled as a stationary zero-mean Gaussian random process. The mean part f D and the fluctuating part f ’D ðtÞ of the drag force per unit height can be stated as follow (Balendra, 1993):
1 2 f D ¼ ρa U CD B ; 2
þkTMD kTMD kTMD kTMD
þcTMD cTMD CTMD ¼ cTMD cTMD
f ’D ðtÞ ¼ ρa UuðtÞCD B
2
SfD ðωÞ ¼ ρ2a U B2 CD 2 Su ðωÞ
(4)
(10)
where Su ðωÞ is the spectral density of the turbulent velocity. Davenport (1961) presented the following formula for the PSD of the longitudinal wind velocity fluctuations for the frequency range of interest of dynamic response of buildings:
where ζs and ζa are structural and aerodynamic damping ratios respectively.
2.2. FRF matrix and the structural responses
Su ðωÞ ¼
The FRF matrix between the generalized structural displacements and the external forces can be obtained using the following equation (Newland, 1996):
600ω π u10
2
2u* 2
2 4=3 jωj ω 1 þ 600 π u10
(11)
pffiffiffi where u* ¼ κu10 denotes the friction velocity; u10 is the reference mean wind speed at 10 m above the ground, and κ is the surface drag coefficient. The correlation of wind velocity fluctuations between two different points is computed using the following formula (Vickery, 1970):
(5)
where ω is the angular frequency and j is the imaginary unit. The response power spectral density (PSD) matrix for an MDOF system is computed as follows (Newland, 1996): Sx ðωÞ ¼ HðωÞSf ðωÞH ðωÞ
(9)
where ρa is the air density, U and u(t) are the mean and the turbulent component of the wind velocity in the along-wind direction; CD ¼ CwþCl is the drag coefficient, Cw and Cl are pressure and suction coefficients; B is the tributary length normal to the wind direction. The power spectral density of the fluctuating force in the along-wind direction is:
where Ks and Cs are stiffness and damping matrices of the building, respectively, and kTMD and cTMD are stiffness and damping constants of the TMD system, respectively. The damping matrix of the structure Cs is obtained using the proportional damping method. The total damping ratio can be obtained as follows:
1 HðωÞ ¼ ω2 M þ jωC þ K
(7)
(1)
(3)
ζt ¼ ζs þ ζa
ω4 Sxii ðωÞdω
where α, a, and b are constants depending on the terrain type, z is the height of interest and Ub is the basic wind velocity. The basic or reference wind velocity in most codes and standards is based on the wind measurements at 10 m height in an open terrain associated with different mean recurrence intervals and averaging times (Zhou et al., 2002).
½0nðn1Þ ½0n2 ½Ks nn ½0n1 þ ; ½01ðn1Þ ½KTMD 22 ½01n 0
∞
There are two kinds of wind-velocity profile descriptions, i.e., the logarithmic and the power law. The general form of the power law is (NBCC, 2010):
where Ms is the mass matrix of the building, and mTMD is TMD mass. The global stiffness and damping matrices can be obtained by assembling the effects of the building and TMD system: K¼
þ∞
3.1. Wind velocity profile
Fig. 2 shows the general configuration of the tapered structure equipped with TMD system, and its analytical model. The global mass matrix of the structure can be written as: ½Ms nn ½01n
Z
3. Wind characteristics
2.1. The effect of the TMD system
E €x2i ¼
and
where Sxii ðωÞ is the ith element on the diagonal of the response PSD matrix.
UðzÞ ¼ aUb
M¼
Sxii ðωÞd ω
8 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > > <jωj cZ 2 zj zk 2 þ cy 2 yj yk 2 = cohjk ðωÞ ¼ exp ðzk Þ > > : 2π ; 1 U z þU
(6)
2
where Sf ðωÞ is the PSD matrix of the excitation forces and * stands for the conjugate transpose. Finally, the displacement and acceleration mean square responses for the ith degree of freedom (DOF) can be obtained as follows:
(12)
j
The suggested values of cy and cz for engineering calculations are 16 and 10, respectively. The aerodynamic damping ratio in the along-wind direction is ob3
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
tained using the following formula proposed by Huilan et al. (2012):
0:5 1 ρa B U H pffiffiffiffiffiffiffi ζs 0:15 CD 0:002 4π ρs D f0 BD B
Table 1 Parameters of the tapered super-tall building with taper ratio.λ ¼ 5%
ζ a ¼ λm
(13)
where ρa =ρs is the ratio of air density to mass density of building, B and D are plan dimensions in the crosswind and along-wind directions, respectively, f0 is the building natural frequency, U is the mean wind velocity, ζs is the structural damping ratio, CD is the drag coefficient. λm is a modification coefficient equal to 1, 0.71, 1.17 and 1.26 for the taper ratios 0, 1%, 3% and 5%, respectively. 3.3. The power spectra of crosswind force In the present study, the following power spectra formula proposed by Liang et al. (2002) is used to compute the crosswind response of the building: nSðnÞ
σ
2
¼A
i 3 HðC1 Þ C 0:5 2 n þ ð1 AÞ 2 2 2 2 2 2 ð1 n Þ þ C1 n 1:56 ð1 n Þ þ C2 n
(14)
in which, σ ¼ 12ρa U 2 ðzÞCL B is the RMS of crosswind force at height z; CL is the mean RMS lift coefficient; B is the breath of windward side; n ¼ n= ns , ns ¼ St UðzÞ=B is the frequency of vortex shedding, St is the Strouhal number; A, C1, C2, and H(C1) are parameters function of the side ratio, height and cross-section area of building. The co-spectra of the crosswind forces between two different levels can be computed using the following coherence function: 2 Δ cohjk ¼ exp ;
α2
Δ¼
zj zk B
AL R5V C 2L R2V ρa B 2 ð1 RV Þ þ SL REVL ρs X
ðRV 1Þ
Height(m)
width(m)
Moment of inertia (m4)
Area (m2)
Mass (ton)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 336 352 368 384 400
49.2 48.4 47.6 46.8 46.0 45.2 44.4 43.6 42.8 42.0 41.2 40.4 39.6 38.8 38.0 37.2 36.4 35.6 34.8 34.0 33.2 32.4 31.6 30.8 30.0
8083 7494 6931 6392 5879 5390 4927 4489 4075 3687 3324 2986 2673 2385 2122 1884 1671 1483 1320 1183 1070 982 920 882 870
787.2 774.4 761.6 748.8 736.0 723.2 710.4 697.6 684.8 672.0 659.2 646.4 633.6 620.8 608.0 595.2 582.4 569.6 556.8 544.0 531.2 518.4 505.6 492.8 240.0
9683 9370 9063 8761 8464 8172 7885 7604 7327 7056 6790 6529 6273 6022 5776 5535 5300 5069 4844 4624 4409 4199 3994 3795 1800
building
height is assumed in the quadratic form: 2 z IðzÞ ¼ ðIbase Itop Þ 1 h þ Itop , where Ibase and Itop are the moment of inertia of the base and top-floor of the example building, respectively. The value of Itop is assumed about 10% of Ibase , and the value of Ibase is determined in such a way that the peak drift index of the building under the assumed wind loading is close to the allowable drift limit 1/600. The peak acceleration is generally taken to be less than 20 mili-g (20 cm/s2) to meet the occupant comfort requirement (Balendra, 1993). The allowable RMS acceleration can be obtained using the general expression pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of peak factor: gp ¼ 2 lnðνTÞ þ 0:5772= 2 lnðνTÞ, where T is the averaging time and ν is the up-crossing rate that is approximately equal to a natural frequency (Hz) of the building (Kwon and Kareem, 2013). For the studied buildings, the peak factor is about 3.75. Therefore, the allowable RMS acceleration is about 5.33 cm/s2. The results show that the obtained crosswind RMS acceleration for the uncontrolled structure is 14.14 cm/s2, which is very higher than the allowable limit and needs to be controlled. The taper ratio is defined as the difference of the top and bottom widths divided by the height of the building. For the tapered tall building, it is assumed that the average value of the top and base widths is equal to the width of the non-tapered building. By this assumption, the bottom width increases by increasing the taper ratio. This leads to increasing the stiffness and decreasing the lateral displacement of the taper buildings. It is assumed that the stiffness of the building is linearly proportional to the bottom width of the building. For instance, for the taper ratio 5%, which is equivalent to the base and top widths 50 and 30 m, respectively, the stiffness matrix of the building is obtained by multiplying that of the non-tapered building by the bottom width ratio 1.25. By this assumption, the results indicate that the top-floor reduced wind velocity, which has an important role in the crosswind response, are equal to each other for the studied buildings. Table 1 presents the structural properties of the typical tapered building with the taper ratio λ ¼ 5% for a 25 D.O.F analytical model, including height, width, moment of inertia, lumped mass, and tributary area of each node. The critical inherent damping ratio for the first mode is assumed ζs1 ¼ 0.01, and the damping ratios for the higher modes are calculated using the following equation (Kareem, 1996):
(15)
in which α2 is a constant, which is equal to 5.56 for the side ratio D/B between 0.25 and 1. The aerodynamic damping ratio in the crosswind direction is obtained using the formula proposed by Gu et al. (2014): ζa ¼
Node
(16)
where RV ¼ Vm =Vs ; Vm is the reduced wind velocity of building [Vm ¼ UH =ðf0 BÞ; Vs is the reduced wind velocity of Strouhal frequency [Vs ¼ UH =ðfS BÞ; B/X is the ratio of building width to the wind-induced displacement response, AL , CL , SL and EL are parameters function of the side ratio, turbulent intensity, aspect ratio and taper ratio. For instance, typical values of AL , CL , SL ; and EL for the building with uniform square cross-section, with aspect ratio 8 and turbulent intensity 11%, are 0.037, 0.89 and 0.202 and 4.62, respectively. 4. Numerical study A numerical study is presented to demonstrate the control effectiveness of combination of tapering method and TMD system. The structures are modeled as a multi-degrees-of-freedom vertical cantilever beam. The PSD matrices of the along-wind and crosswind loads are obtained using the equations presented in Sect. 3, and the frequency-domain analysis, described in Sect. 2, is used to compute the along-wind and crosswind RMS displacement and acceleration values. The effect of different values of taper ratios and TMD mass on the control performance of the structure is investigated. The physical properties of the example super-tall building and the assumed aerodynamic information are presented in the following paragraphs. The super-tall building is assumed with the height H ¼ 400m and square cross-section. Width and depth for non-tapered building were considered 40m. The bulk mass density of the building is assumed ρs ¼ 250 kg/m3. The variation of the moment of inertia along the 4
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
Fig. 3. Variation of the top-floor along-wind and crosswind responses versus basic wind speed: (a) RMS displacement, (b) RMS acceleration.
fi ζ si ¼ ζs1 1 þ 0:38 1 f1
(17)
Table 2 Optimum TMD parameters and the corresponding along-wind and crosswind responses.
where f1 and fi are the first ant ith natural frequency of the structure. The aerodynamic physical data are chosen as follows: air density ρa ¼ 1.25 kg/m3; 1-h average basic wind speed Ub ¼ 30 m/s, which is equivalent to 3-sec gust wind speed of 47.4 m/s; gradient height zg ¼ 383 m; exponent for mean wind power α ¼ 0.36. The drag coefficients is assumed CD ¼ 1.3 as proposed by Canadian code (NBCC, 2010) for tall buildings with square cross-section. The surface drag coefficient is assumed κ ¼ 0.03. The crosswind PSD matrix parameters appeared in Sect. 3.3 are obtained as follows: A ¼ 0.96, C1 ¼ 0.019, C2 ¼ 2.0, H(C1) ¼ 0.093, α2 ¼ 5:56, the mean RMS lift coefficient CL ¼ 0.404, Strouhal number St ¼ 0.084 (Liang et al., 2002).
TMD mass Ratio μn
0.0% 0.4% 0.8% 1.2% 1.6% 2.0%
Using the procedure described in Sections 3 and 4, the along-wind and crosswind responses are obtained for the studied structures presented in Fig. 1. The assumptions and the results for each structural condition are presented in the following sections. Since the greatest wind-induced responses occur at the top-floor of the structure, the topfloor RMS displacement and acceleration values are presented and discussed herein. In order to evaluate and compare the control performance of the structures, the control effectiveness is defined as the percentage reduction of the top-floor RMS response compared to that response of the non-tapered building, which can be stated as follows:
fdopt
σ xa (cm)
0.0385 0.0541 0.0659 0.0756 0.0841
0.996 0.992 0.988 0.984 0.980
σ x€a (cm/
σ xc (cm)
s2) 7.74 6.34 6.01 5.82 5.69 5.59
7.22 5.37 4.89 4.60 4.40 4.24
σ x€c (cm/ s2)
15.54 13.15 12.55 12.21 11.99 11.83
14.14 10.84 9.91 9.35 8.95 8.65
In this section, wind-induced responses are presented for the example super-tall building equipped with TMD systems having different values of mass. The frequency ratio fd and the damping ratio ζd for the TMD are defined as:
(18)
where σ u and σ c are the top-floor RMS responses of the uncontrolled and controlled structures, respectively.
fd ¼ ωd =ωn
5.1. Non-tapered super-tall building (uncontrolled structure)
and
ζd ¼ cd =ð2md ωn Þ
(20)
where md , cd , and ωd are the mass, damping, and frequency of the TMD system, and ωn is the lowest natural frequency of the building. The TMD tuning formulas presented by Hahnkamm, 1932 and Brock, 1946 are used to optimize the response of the building:
Fig. 3(a) and (b) show the variation of the top-floor along-wind and crosswind RMS displacement and acceleration responses, respectively, versus basic wind speed. These figures show that the crosswind responses are higher than the along-wind responses. For the presented values of the basic wind speed, difference between the crosswind and along-wind responses increases as the wind speed increases. The crosswind force is due predominantly to the fluctuating lift force induced by the vortices shed in the building’s wake (Simiu, 2011). The resonant or critical wind velocity, at which the vortex shedding frequency coincides with the natural frequency of the building, can be obtained using the following equation: Ucr ref ¼ Bref f 0 St
ζd opt
Crosswind
5.2. Non-tapered super-tall building with TMD system
σu σc *100 σu
Along-wind
equal to 0.084 for the square cross-section (Liang et al., 2002), and f 0 is the dominant frequency of the structure equal to 0.166 for the non-tapered super-tall building. According to this equation, the critical top-floor wind speed for the studied non-tapered building is about 79 m/s. For the basic wind speed Ub ¼ 30 m/s, the top-floor wind velocity from the power law is UH ¼ 47.4 m/s, which is lower than the resonant wind velocity as expected. For this condition, the top-floor responses are as follows: the static displacement 22.22 cm, the along-wind and crosswind RMS displacements σ xa ¼ 7.74 and σ xc ¼ 15.54 cm, respectively, and the along-wind and crosswind RMS accelerations σ x€a ¼ 7.22 and σ x€c ¼ 14.14 cm/s2, respectively. This result shows that for the example
5. Analysis and results
Control effectiveness ¼
Optimum parameters
fd opt ¼
1 1þμ
and
ζ dopt ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3μ 8ð1 þ μÞ3
(21)
where μ is the mass ratio of TMD to the generalized mass of first mode of the building. Table 2 presents the optimum TMD parameters and the corresponding optimized along-wind and crosswind RMS displacements σ xa and σ xc , and RMS accelerations σ x€a and σ x€c , respectively. The TMD mass ratio μn is defined as the mass ratio of TMD to the primary modal mass of the nontapered tall building. The maximum value of μn is assumed 2%, which is
(19)
where Bref is the reference crosswind width, St is the Strouhal number 5
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
The first natural frequency of the tapered buildings are presented in Table 3 for different values of λ. As it is expected, the natural frequency of the structure increases as the taper ratio increases due to decreasing the effective modal mass and increasing the stiffness of the tapered buildings. The acceleration response is directly proportional to the displacement response and square of the natural frequency. Therefore, the acceleration response does not show considerable change versus variation of taper ratio due to increasing the natural frequency. Both along-wind and crosswind acceleration responses increase for taper ratio λ ¼ 1% compared to the corresponding responses of non-tapered building. The along-wind acceleration in the range λ ¼ 1% to 5% decreases by increasing the taper ratio. However, the reduction percentage is considerably lower than that of the displacement response. For instance, the along-wind acceleration reduction for λ ¼ 5% is 5.4%. The crosswind accelerations of tapered building due to lower aerodynamic damping are higher than that of non-tapered building. Fig. 5 shows the variation of the crosswind acceleration and displacement reduction percentages versus the taper ratio. This figure shows that the aerodynamic modification of tapering along the building height can effectively reduce displacement of the structure. However, the acceleration response of tapered building show 0–5.5% increase compared to that response of non-tapered building, depending on the taper ratio. The maximum increase is related to the taper ratio λ ¼ 1%. Therefore, tapering method is not effective in controlling the acceleration response.
Fig. 4. Variation of the control effectiveness of the crosswind acceleration and displacement versus μn .
equivalent to a 686-ton TMD system. The results show that the control effectiveness of the TMD system increases as the TMD mass increases for both along-wind and crosswind responses. For μn ¼ 2%, the along-wind and crosswind RMS displacements decrease by 27.8% and 23.9%, respectively, compared to the corresponding responses of the uncontrolled structure. The along-wind and crosswind response reduction values for the RMS accelerations are 41.3% and 38.8%, respectively. This result indicates that the control effectiveness of the TMD system for the along-wind and crosswind responses are approximately close to each other. Fig. 4 shows the variation of the crosswind acceleration and displacement reduction percentages versus TMD mass ratio μ. This figure indicates that the control effectiveness of the TMD system for the acceleration is higher than that for the displacement. For example, the crosswind acceleration and displacement reduction percentages for μn ¼ 2% are 38.8 and 23.9%, respectively. This figure also shows that the control effectiveness of the TMD system increases with increasing μn , while the slope of the variation curve decreases. For instance, the absolute value of the slope of the acceleration curve at μn ¼ 0:3% is about 7 times of that at μn ¼ 2%. This means that the higher control efficiencies of the TMD system require using very heavier mass dampers.
5.4. Tapered super-tall building with TMD system The control effectiveness of the combination of tapering method and TMD system for different values of TMD mass ratio and taper ratio are presented herein. Fig. 6 shows the variation of the crosswind displacement and acceleration reduction percentages for the tapered super-tall building with a TMD system with mass ratio μn ¼ 2% versus taper ratio. This figure indicates that using the control techniques of tapering
5.3. Tapered super-tall building without TMD system Wind-induced responses for the tapered super-tall building for different values of taper ratio λ between zero and 5% are presented and discussed in this section. Taper ratio is defined as the difference between the top and bottom widths divided by the height of the building. Table 3 presents the top-floor along-wind and crosswind RMS responses, as well as, the first natural frequency of the super-tall buildings and the alongwind and crosswind aerodynamic damping ratios for different values of λ. The results show that the along-wind and crosswind displacement responses decrease with increasing λ. For λ ¼ 5%, the along-wind and crosswind RMS displacement decrease by 42.6% and 41.7% compared to the corresponding responses of the non-tapered building, respectively.
Fig. 5. Variation of the control effectiveness of the crosswind acceleration and displacement versus.λ
Table 3 Top-floor along-wind and crosswind responses versus taper ratio for the tapered buildings. Taper ratio λ
0 1% 2% 3% 4% 5%
Natural frequency
0.166 0.176 0.186 0.197 0.209 0.221
Along-wind
Crosswind
ζ a ð%Þ
σ xa (cm)
σ x€a (cm/s )
ζ a ð%Þ
σ xc (cm)
σ x€c (cm/s2)
0.39 0.27 0.36 0.44 0.46 0.49
7.74 7.13 6.26 5.52 4.95 4.44
7.22 7.49 7.21 6.95 6.88 6.80
0.55 0.32 0.33 0.31 0.27 0.19
15.54 14.41 12.67 11.24 10.03 9.06
14.14 14.92 14.59 14.43 14.39 14.59
2
6
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
structure increases with increasing μn and λ. The maximum control effectiveness, which can be achieved for μn ¼ 2% and λ ¼ 5%, are 58.24 and 46.18% for the displacement and acceleration responses, respectively. Table 4 presents the crosswind RMS displacement and acceleration values and the corresponding reduction percentages for different structural condition including non-tapered building, tapered building (λ ¼ 5%), non-tapered building with TMD (μn ¼ 2%), and tapered building equipped with TMD (λ ¼ 5%, μn ¼ 2% ). This table also shows the average reduction percentages of displacement and acceleration responses for each case. As we can see, the average response reduction for the tapered building without TMD is 19.26%, which increases to 52.21% when the TMD system is used. This result indicates that using the control strategies of tapering and TMD system together reduces more efficiently both acceleration and displacement responses compared to the conditions that only one of the control methods is used. This method can be very helpful in satisfying the occupant comfort criteria and safety requirements in design of super-tall buildings under wind loads.
Fig. 6. Variation of the reduction percentages of the crosswind acceleration and displacement for a tapered building with a TMD system for μn ¼ 2% versus taper ratio.λ
6. Conclusions
and TMD together can be effective in reducing both displacement and acceleration responses. The value μn ¼ 2% is equivalent to a 686-ton TMD system. By increasing the taper ratio, the primary modal mass of the structure decreases due to decreasing the volume of tapered building by height. Therefore, for a constant value of TMD mass, the effective mass ratio of the TMD increases that results in increasing the control effectiveness of the TMD system for taper buildings. The displacement and acceleration reduction curves intersect at λ ¼ 2.25%, which results in about 42% displacement and acceleration reduction. For λ ¼ 5%, the displacement and acceleration reduction values are 58.24% and 46.18%, respectively. Figs. 7 and 8 show the contour plot of reduction percentages of crosswind displacement and acceleration, respectively, versus different values of TMD mass ratio μn and taper ratio λ. The maximum values of μn and λ are assumed 2% and 5%, respectively. For every pair of μn and λ, the corresponding response reduction percentage can be obtained using the contour plots. For instance, for μn ¼ 1% and λ ¼ 3%, the reduction percentages of crosswind displacement and acceleration are 45 and 36%, respectively. These figures show that the control effectiveness of the
Tapering of building along the height has been studied as an aerodynamic modification method to reduce wind-induced vibrations. As reported in the previous studies, although this control method can effectively reduce the wind-induced drift of tall buildings, it is not efficient in reduction of the acceleration response. In the present study, tapering method and TMD system are used together to achieve higher control performance for the acceleration and displacement responses. In order to investigate the control performance of the studied systems, a numerical example of a 400-m high super-tall building is presented, and the responses are computed for the following conditions: (a) non-tapered building; (b) non-tapered building with TMD system; (c) tapered building and (d) tapered building with TMD system. The TMD mass ratio μn is defined as the mass ratio of TMD to the primary modal mass of the nontapered tall building. The maximum taper ratio and TMD mass ratio are assumed λ ¼ 5% and μn ¼ 2%, respectively, and the crosswind and alongwind responses are computed for different values of λ and μn using the frequency domain analysis of MDOF structures. For the controlled structures, the control effectiveness is defined as the percentage reduction in the building top-floor RMS response compared to that of the un-
Fig. 7. Contour plot of reduction percentages of the crosswind displacement versus different values of μn and λ. 7
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Journal of Wind Engineering & Industrial Aerodynamics 196 (2020) 104031
Fig. 8. Contour plot of reduction percentages of the crosswind acceleration versus different values of μn and λ.
Table 4 The crosswind RMS displacement and acceleration values and the corresponding reduction percentages for different structural condition. Structural condition
Non-tapered Tapered (λ ¼ 5%) Non-tapered with TMD (μn ¼ 2%) Tapered with TMD (λ ¼ 5%, μn ¼ 2% )
Crosswind
Control effectiveness (%)
σ xc (cm)
σ x€c (cm/s )
Displacement
Acceleration
15.54 9.06 11.83 6.49
14.14 14.59 8.65 7.61
0 41.70 23.87 58.24
0 3.18 38.83 46.18
2
Average reduction (%)
0 19.26 31.35 52.21
Funding
controlled structure. The results indicate that for the studied uncontrolled super-tall building with the assumed structural and wind characteristics, the crosswind response is higher than the occupant comfort level and needs to be controlled. The first natural frequency of the tapered super-tall buildings increases with increasing λ, due to decreasing the effective modal mass and increasing the stiffness of the structure. For instance, the natural frequency of the building with λ ¼ 5% is 33% more than that of the non-tapered building. Since the acceleration response is proportional to the displacement response and square of the natural frequency, although tapering effect is efficient in reduction of displacement, it cannot effectively reduce the acceleration response due to increase of the natural frequency. For instance, the displacement reduction percent for the taper ratio λ ¼ 5% is 41.7%, while the acceleration response increased by 3.18%. For the non-tapered super-tall building equipped with TMD system, the control effectiveness of acceleration is higher than that of the displacement, and the vibration control effect increases as the TMD mass ratio increases. The displacement and acceleration control effectiveness for the non-tapered building with TMD mass ratio μn ¼ 2% are 23.87 and 38.83%, respectively. The highest control performance is observed for the tapered building with TMD system. For instance, for the structure with μn ¼ 2% and λ ¼ 5%, the displacement and acceleration control effectiveness are 58.24 and 46.18%, respectively. The average displacement and acceleration control effectiveness for the conditions including tapered λ ¼ 5%, non-tapered equipped with TMD μn ¼ 2%, and tapered equipped with TMD λ ¼ 5% and μn ¼ 2% are 19.26, 31.35 and 52.21%, respectively. These results demonstrate the efficient performance of using tapering method and TMD system together in reducing the wind-induced vibration of super-tall buildings, which can satisfy the safety and occupant comfort criteria under wind loads.
Funding was received for this work. All of the sources of funding for the work described in this publication are acknowledged below: Financial support from Iran National Science Foundation (INSF) through Grant. 97011187. Declaration of competing interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. Acknowledgments The author is grateful for the financial support from Iran National Science Foundation (INSF) through Grant. 97011187. References Amano, T., 1995. The effect of corner-cutting of three dimensional square cylinders on vortex-induced oscillation and galloping in uniform flow. J. Struct. Constr. Eng., AIJ 60 (478), 63–69. Amin, J.A., Ahuja, A.K., 2010. Aerodynamic modifications to the shape of the buildings: a review of the state-of-the-art. Asian J. Civ. Eng. 11 (4), 433–450. Asami, T., Nishihara, O., Amr, M., 2002. Analytical solutions to H∞ and H2 optimization of dynamic vibration absorbers attached to damped linear systems. J. Vib. Acoust. 124 (2), 284–295. Balendra, T., 1993. Vibration of Building to Wind and Earthquake Loads. Springer-Verlag, London. Brock, J.E., 1946. A note on the damped vibration absorber. ASME J. Appl. Mech. 13 (4), A-284. 8
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