Effects of Amplitude-Dependent Damping and Time Constant on Wind

Computers and Structures 85 (2007) 1165–1176 www.elsevier.com/locate/compstruc

Eﬀects of amplitude-dependent damping and time constant on wind-induced responses of super tall building

J.R. Wu a,b, P.F. Liu b, Q.S. Li b,*

a Department of Civil Engineering, Jinan University, Guangzhou 510632, China b Department of Building and Construction, City University of Hong Kong, Hong Kong, China

Received 10 January 2007; accepted 11 January 2007 Available online 6 March 2007

Abstract

Full-scale measurements of wind-induced responses of a 79-story tall building, Di Wang Tower, were conducted during the passages of several typhoons. The amplitude-dependent damping ratios of the super tall building were obtained from the measurements. A Monte Carlo simulation procedure was developed in this study to generate fluctuating along-wind and across-wind forces acting on this building. The wind-induced responses of Di Wang Tower were numerically evaluated in time domain on the basis of the generated fluctuating wind forces and the established finite element model of the building. The predicted dynamic responses of the building using the actual amplitude-dependent damping characteristics were compared to those computed with constant damping parameters assumed by the structural designers to evaluate the adequacy of current design practices and to investigate the effect of amplitude-dependent damping on the wind-induced responses. Finally, the effect of time constant on the wind-induced responses of Di Wang Tower was studied by comparing the time domain computational results with those from conventional spectral analysis method. Some of the research findings resulted from this combined experimental and numerical study are expected to be of interest and practical use to professionals and researchers involved in the design and analysis of super tall buildings. 2007 Elsevier Ltd. All rights reserved.

Keywords: Tall building; Damping; Wind eﬀect; Typhoon; Dynamic response; Full-scale measurement

1. Introduction (transverse) are 68.55 m and 35.5 m, respectively. There- fore, the aspect ratio between height and transverse width Di Wang Tower is located at northwest of downtown is about 9, which means that it largely exceeds the relevant Shenzhen, Guangdong province, China, including 68- criteria laid down in the current design codes and standards storey main office tower, plus 11-storey facility and refuge in China. This implies that this tall building is a slender floors as well as top tower structures, as shown in Fig. 1a. structure. As Shenzhen is located at the edge of the most Total, the main structure of the tall building is 79-story active typhoon generating area in the world, this super tall and is about 325 m high from ground level. There are two building may be susceptible to severe vibration induced by masts with 59 m high erected on the roof of the tower. typhoons. Therefore, there is a need for investigating its The height from the ground to the top of the masts is about dynamic performance under typhoon conditions. 384 m. Di Wang Tower was the tallest building in Mainland Natural frequencies and damping ratios are very impor- China when it was built several years ago. The lengths of the tant parameters which affect the dynamic response of struc- main building in X-direction (longitudinal) and Y-direction tures under dynamic actions such as wind or earthquake excitation. The natural frequencies can be conveniently determined from conventional methods with reasonable * Corresponding author. Tel.: +852 27844677; fax: +852 27887612. accuracy. However, it is very difficult or impossible to E-mail address: [email protected] (Q.S. Li). determine structural damping ratio accurately prior to

istic value in dynamic analysis either in time domain or frequency domain. Li et al. [14] presented the first study in wind engineering to investigate the effects of amplitude- dependent damping on wind-induced vibration of a tall building in Hong Kong. However, in their study, the damping measurements were made under moderate typhoon conditions and the tall building was simplified as a five lumped mass system in the dynamic analysis. Obvi- ously, it would be more meaningful to investigate such effects under larger vibration amplitude levels and a tall building should be modeled as a 3D multi-degree of freedom system. Therefore, further comprehensive studies are required on this research topic. In this study, the damping database was accumulated from field measurements of wind effects on Di Wang Tower during the passages of Typhoons Sally, York, Imbudo, Dujuan and Compass. It was reported by the Hong Kong Observatory that Typhoon York was the strongest typhoon to hit Hong Kong and Shenzhen since 1983 and Typhoon Dujuan was the strongest typhoon to attack the Pearl River Delta region since 1979. This thus allowed us to obtain the damping data from the super tall building under severe vibration amplitudes induced by the strong typhoons, which provides a basis for further study on the effects of amplitude-dependent damping on the dynamic responses of the building. The product of the natural frequency and damping ratio, which is related with time constant of a structure, indicates how long the structure takes to response to an action [10]. When a time constant becomes large then the response of a structure to wind action may be delayed. The reduction factor of root mean square (RMS) displacement response of a structure with long time constant can be as high as 20% compared with the results from conventional spectral analysis method [10]. The effect of time constant on the reduction of dynamic response of a tall building was studied by Jeary [10]. However, the structural Fig. 1a. Overview of Di Wang Tower. system of the building was simplified as a single degree of freedom system and only a linear sway mode was considered in his study. As discussed previously, it is desirable construction. Structural damping ratio is usually assumed to model a tall building as a multi-degree of freedom sys- to be constant value at design stage. However, the actual tem and include more numbers of modes in the dynamic damping ratio is found to be a nonlinear parameter with analysis for investigating the time constant effect on the amplitude-dependent property [9,28,12,13,17–19]. Little dynamic responses of tall buildings. work has been done to examine and validate the assump- This study, taking Di Wang Tower as an example, will tions made on structural damping of tall buildings. Over address two important issues in dynamic analysis of tall the last three decades, significant measurements of struc- buildings: the effects of amplitude-dependent damping tural damping have been made throughout the world and time constant on the wind-induced dynamics response [9,28,23]. However, the majority of the previous measure- of super tall building. First of all, the amplitude-dependent ments were made for buildings with 10–50 stories. There damping ratios are obtained on the basis of the full scale is a serious scarcity of damping data measured from super measurements of wind-induced vibrations of Di Wang tall buildings (building height > 300 m), especially under Tower during several typhoons. Then an empirical ampli- strong wind excitations. Therefore, there is a pressing need tude-dependent damping model is established from the to collect such a database. accumulated damping data. At the second stage, the time Wind-induced responses of a tall building can be esti- histories of fluctuating along-wind and across-wind forces mated in time domain [30,20,11] or frequency domain are generated by the Monte Carlo simulation procedure [5,22]. The damping ratio is usually treated as a determin- developed from the weighted amplitude wave superposition J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176 1167

(WAWS) method [24,6]. The finite element model of the outer frame are filled with C45 concrete to increase the col- building is established for the purpose of conducting the umn stiffness. The basic plan form of the tower is essen- dynamic analysis of the building. The along-wind and tially rectangular with two semi-circle of 12.5 m radius on across-wind responses of this building are numerically two sides, as shown in Fig. 1b. Detailed descriptions of evaluated in time domain and the effects of amplitude- the structural system of this tall building are given in Li dependent damping on the wind-induced responses are and Wu [15]. investigated through comparison between the results predicted using the actual amplitude-dependent damping characteristics and those computed by using constant damping 3. Field measurements of amplitude-dependent damping parameters assumed by the structural designers. Finally, the effect of time constant on the along-wind response of Full-scale measurements of wind effects on Di Wang the building is studied by comparing the time domain com- Tower were conducted by Li et al. [16] during the passage putational results with those from conventional spectral of Typhoon Sally in September, 1996. Xu et al. [32] carried analysis method. The objective of this combined experi- out the field measurements of the wind-induced responses mental and numerical study is to further the understanding of the tall building during a strong typhoon (Typhoon of the effects of amplitude-dependent damping and time York) on 16 September 1999. Continuous efforts have been constant on the wind-induced dynamic responses in order made by the authors to monitor the typhoons effects on to evaluate the adequacy of current design practices and this super tall building since 2002, including the measure- to apply that knowledge to the structural design of super ments made during the passages of Typhoons Imbudo tall buildings. and Dujuan in 2003 [31] and Typhoon Compass in 2004. These field measurements all involved at least two accelerometers installed at the 68th floor of the main tower along 2. Structural system of Di Wang Tower X- and Y-direction to measure the acceleration responses of the building, as shown in Fig. 1b. Significant field data The structural system of Di Wang Tower utilises both have been recorded over the last several years. The ampli- steel and reinforced concrete (SRC), including core wall tude-dependent damping ratios of the building were deter- systems and perimeter steel frame coupled by outrigger mined by the random decrement technique based on the trusses at four levels (second, 22nd, 41st and 66th floors). measured acceleration response data. With the accumu- Besides this, two rows of the vertical bracing are arranged lated measurement results and damping data, an empirical along the building height. The box-type steel columns at model of amplitude-dependent damping will be proposed in this paper. According to a damping model developed by Li et al. [12,13] for tall buildings, it includes two parts: (1) in low amplitude region: the damping ratio is determined as a nonlinear function of amplitude of vibration response within a certain range; (2) in high amplitude plateau: the damping ratio is a constant when the amplitude of vibration response is in a higher level. As the damping characteristics of the tall building are known from the field measurements (see Figs. 2–5) made in Typhoon Sally [16], Typhoon York [32] and from our recent results measured in Typhoons Imbudo, Dujuan and Compass, the amplitude-dependent damping ratios are modeled below: ( 3:1 104x þ 0:0043 0 6 x 6 82:90 mm nNLY1 ¼ ð1Þ 3% 82:90 mm < x

-3 x 10

10 y=0.00031*x+0.0043

8 York Sally 6 Dujuan Imbudo Damping ratio Damping 4 Fitted

0 2 4 6 8 10 12 14 16 18 20 Amplitude (mm) Fig. 1b. Plan of a standard floor, location of the accelerometers and definitions of wind attack angle. Fig. 2. The measured damping data for the first mode in Y-direction. 1168 J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176

x 10-3 and shown in these ﬁgures comprise both structural damp- 7 y=0.000055*x+0.00499 ing and aerodynamic damping. For a steel-concrete com- 6 posite structure such as Di Wang Tower, a constant of York 5 Sally damping value of 2–3% is widely adopted in its structural Compass 4 Dujuan design according to many current design codes and stan- Damping ratio Imbudo Fitted dards. Therefore, the upper plateau deﬁned in the proposed 3 0 2 4 6 8 10 12 14 16 18 20 damping model is taken as a constant value of 0.03. Amplitude (mm)

Fig. 3. The measured damping data for the second mode in Y-direction. 4. Generation of ﬂuctuating wind forces on the building

-3 x 10 The simulation of stochastic variables with arbitrary 10 power spectral representation has been studied extensively. 8 y=0.00088153*x+0.004293 York Shinozuka and Jan [24], Shinozuka [25] developed efficient Sally methods for digital simulation for one-dimensional multi- 6 Compass Dujuan variable stochastic process by a series of cosine functions 4 Imbudo Damping ratio Fitted with weighted amplitude, almost evenly spaced frequencies 2 0 1 2 3 4 5 6 7 and random phase angles. This method, also termed as Amplitude (mm) weighted amplitude wave superposition (WAWS) method Fig. 4. The measured damping data for the first mode in X-direction. [6], has an attractive feature that fast Fourier transform (FFT) can be adopted in the simulation procedure. It has been widely used for generating multi-variate random vari- -3 x 10 10 ables with specific spectral representation. Therefore, it is adopted in this study to generate the fluctuating along- 8 y=0.00088153*x+0.004293 York wind and across-wind forces acting on Di Wang Tower. Sally 6 Compass Dujuan 4 Imbudo Damping ratio 4.1. Simulation of along-wind fluctuating wind forces Fitted 2 0 1 2 3 4 5 6 7 Amplitude (mm) Based on the established 3D finite element (FE) model of Di Wang Tower, the lateral condensed stiffness and mass Fig. 5. The measured damping data for the second mode in X-direction. matrices in X- and Y-directions for this building are formed ( for the analysis of wind-induced dynamic responses. In 0:55 104x þ 0:00499 0 6 x 6 454:72 mm each direction, the structural system of this 79-storey tall nNLY2 ¼ building can be considered to be an equivalent cantilever 3% 454:72 mm < x structure with 79 lumped masses. The spectral density of ð2Þ ( the fluctuating along-wind drag force on each lumped mass 8:81 104x þ 0:004293 0 6 x 6 29:16 mm can be expressed as follows: nNLX1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3% 29:16 mm < x 2 SFDijðzi; zj; f Þ¼ðqCDÞ AziAzjV ziV zj SV ðzi; f ÞSV ðzj; f Þ ð3Þ ( Cohðzi; zj; f Þð5Þ 8:57 104x þ 0:0043 0 6 x 6 29:97 mm nNLX2 ¼ ð4Þ where q is the air density, C is the drag coefficient. A , V 3% 29:97 mm < x D zi zi are tributary area and mean wind speed at height zi, respectively. SV (zi,f ) is the auto-power spectral density of the where nNLX1 is amplitude-dependent damping ratio for the along-wind fluctuating wind speed, which is represented first vibration mode in X-direction, the other three param- by the widely-used von-Karman spectrum with the follow- eters nNLX2,nNLY1,nNLY2 are defined in the similar way. It ing form [26]: was shown from Eqs. (1)–(4) that the damping ratios in X- fL fS ðz ; f Þ 4 V direction are normally larger than those in Y-direction at V i V zi 2 ¼ 5=6 ð6Þ the same vibration amplitude level and the damping ratio rV 2 zi 1 þ 70:8 f LV for the second vibration mode in Y-direction is found to V zi increases more slowly with increasing amplitude.

The obtained damping ratio curves (damping ratio (%) where rV zi and LV are the RMS value of fluctuating wind against response amplitude) shown in Figs. 2–5 clearly speed and turbulence integral length scale at height zi, demonstrates amplitude-dependent energy dissipation respectively. The coherence function Coh(zi,zj,f) of fluctu- characteristics of the building, ie., damping increases with ating wind speed is represented by the following equation increasing amplitude. The values of damping estimated [30]: J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176 1169 Dz height; (2) there is a lack of information on correlations Cohðz ; z ; f Þ¼exp 8 f ð7Þ i j V between across-wind dynamic loads at different building heights; (3) the fundamental mode shape of a tall building V ziþV zj where Dz = jzi zjj and V ¼ 2 . Normally the equation is assumed to be linear. Another available way to establish SFDij(zi,zj,f)=SFDji(zj,zi,f) is always satisfied. Thus, it im- the empirical formulas is to make simultaneous pressure plies that SFD is a real symmetric matrix. The Choleskey measurements from rigid model surfaces, which can not decomposition to the real symmetric matrix SFD can be only determine the integral fluctuating wind forces on a T thus performed by the following form: SFD = HH . The building model but also provide information on the spatial simulated fluctuating along-wind load on the ith lumped and temporal distributions of wind loads over the surfaces mass can be expressed in the following equation [24,25]: of the building. It is thus decided to adopt the empirical Xi XN pffiffiffiffiffiffiffiffiffiffiffi spectral models [21] established from pressure measure- F Dðzi; tÞ¼ jH ijðzi; zj; xkÞj 2Dxk½cosðxkt þ wjkÞ; ments on rigid models as the target spectral representation j¼1 k¼1 for generating fluctuating across-wind forces acting on Di i ¼ 1; 2; ...; M ð8Þ Wang Tower. The co-spectral density for the fluctuating across-wind forces along the building height is expressed as where Dxk is frequency interval, M is the number of 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lumped masses representing that of the floors of the build- 1 2 2 Sfyðzi; zj; f Þ¼ qCL AziAzjV ziV zj fy ðzi; f Þfyðzj; f Þ ing, N is the number of frequency intervals, xk = 2 (k 1)Dx and w is independent random phase angle uni- k jk Cohðzi; zj; f Þð9Þ formly distributed between 0 and 2p. Hij(zi,zj,xk) is the corresponding entry in Choleskey decomposed matrix H. where CL is the RMS lift coefficient, Azi,Azj are the areas According to the suggestion given by Holmes [8], the equiv- corresponding to the ith and jth lumped mass and Sfy ðzi;f Þ alent full-scale upper frequency response limit for wind- fyðzi; f Þ¼ 2 is the normalized power spectral den- 1qV 2 C B induced pressure measurements should not be less than ðÞ2 zi L 2 Hz. Therefore, for generating wind loads by Eq. (8), the sity of the fluctuating across-wind force at height zi, which cut-off upper circular frequency is selected to be 4p rad/s can be represented by the following equations [21]: 2 0:5 2 and the number of frequency intervals is taken to be 512, fS ðzi; f Þ HðC Þn C n fy ¼ A 1 þð1 AÞ 2 and the time-step for the wind loading simulation is 0.05 s. 2 2 2 2 2 2 2 r ð1 n Þ þ C1n 1:56½ð1 n Þ þ 2n 1 6 4.2. Simulation of fluctuating across-wind forces for D=B < 3 ð10Þ 4 ÀÁ 2 0:5 n 3 fS ðzi; f Þ 1:275 n C fy ¼ A þð1 AÞ 2 k It is well known that the mechanisms of across-wind 2 2 2 2 2 r 1 n C n 2 C n2 ð Þ þ 1 1:56 1 n 2 dynamic loads on tall buildings are much more complex k2 þ k2 than those of along-wind dynamic loads. It has been recog- for 3 6 D=B 6 4 ð11Þ nized [22] that across-wind dynamic loads on tall buildings are induced by three major mechanisms: along-wind turbu- The normalized coherence function Coh(zi,zj,f) for the lence, across-wind turbulence and wake excitation, with fluctuating across-wind forces has the following form: "# wake excitation being the main contributor. However, 2 due to non-homogeneous pressure fluctuations in sepa- D jzi zjj Cohðzi; zj; f Þ¼cosða1DÞ exp ; D ¼ rated and wake regions, it is very difficult to establish an a2 B analytic model for estimation of across-wind dynamic ð12Þ loads on tall buildings. Therefore, strictly speaking, accu- rate analytical calculation methods to evaluate across-wind All the parameters involved in Eqs. (10)–(12) are given in dynamic responses of tall buildings are not available in Liang et al. [21]. Once the target power spectral density is literatures, while several procedures, e.g., gust factor determined, the fluctuating across-wind forces acting on approach [5], have been developed for predicting loads each lumped mass of the building can be generated by the and response in along-wind direction. Wind tunnel tests same simulation procedure as described in the last section. are usually needed to evaluate the across-wind loads on tall buildings. Based on numerous wind tunnel tests for rectan- 5. Time history dynamics analysis of the wind-induced gular tall building models with different aspect and side response of Di Wang Tower ratios, several empirical formulas for the estimation of power spectral density of fluctuating across-wind force The equation of motion of this building in X-orY-direc- were proposed by previous researchers based on the mea- tion can be expressed in the following equation: surements of across-wind loads by the high frequency force ½Mf€ygþ½Cfy_gþ½Kfyg¼fFðtÞg ð13Þ balance technique [1,3,27]. However, such a technique may have three shortcomings: (1) there is no information on where [M], [C]and[K] are the 79 · 79 order mass, damping variations of across-wind dynamic loads along building and lateral condensed stiffness matrices of the structural 1170 J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176 system in X-orY-direction. f€yg and {F(t)} are vectors of results of the wind-induced responses are presented and dis- acceleration response and fluctuating wind force, respec- cussed in the following sections. tively. The damping matrix of the building was determined based on the Rayleigh damping model [4], in which the 5.1. The along-wind responses (in X-direction) under damping matrix is assumed to be proportional to the mass different load cases and stiffness matrices of the structural system, i.e. ½C¼a½Mþb½Kð14Þ The time histories of the along-wind acceleration and displacement responses at the top floor of the building where a, b are the proportional constants related to the dy- are shown in Figs. 6 and 7 with consideration of several namic properties of the structure, which can be determined constant damping ratios and the proposed amplitude- from the damping ratios n1, n2 and natural frequencies x1, dependent damping model when the mean wind speed atop x2 of the first two vibration modes, respectively [4]. Since the building is 15 m/s. The wind-induced responses during Eq. (13) is a nonlinear equation owing to the amplitude- the first 3-min are shown in these figures for demonstration dependent matrix C or f, for obtaining the time domain purposes. It can be seen from these figures that the wind- solution of Eq. (13) the step-by-step Newmark integration induced responses based on the proposed amplitude-depen- method [4,29] is used in this study. Iteration algorithms are dent damping model are greater than those for the cases of needed to consider the amplitude-dependent damping effect constant damping ratios 1% and 3%. The RMS and maxi- on the dynamic responses.

The time history analysis for the wind-induced response Acceleration for constant damping ratio 1.0% 0.8 of this building in X- and Y-direction was conducted with Acceleration for amplitude-dependent damping ratio consideration of different mean wind speeds atop the build- 0.6 ing. The building was assumed to be an isolated building located in Terrain type D as specified in the design load 0.4 code of China (GBJ 9-87) [2]. The empirical amplitude- 0.2 dependent model proposed in this study for Di Wang 0 Tower was adopted in the computation. Meanwhile, constant damping ratios of 1%, 2% and 3% (such as what were Acceleration (gal) -0.2 assumed by the structural designers in the design stage of -0.4 this building) were also considered in this study. Therefore, comparisons will be made between the responses predicted -0.6 by using the measured actual damping characteristics and 0 20 40 60 80 100 120 140 160 180 those computed by using the constant values of damping Time (s) to examine the effect of amplitude-dependent damping on Fig. 6. Along-wind acceleration in X-direction atop the building (mean the dynamic responses of this super tall building. According wind speed = 15 m/s). to the design code of China (GBJ 9-87) [2] and Li et al. [17], the design mean wind speeds corresponding to different Acceleration for constant damping ratio 3.0% 0.8 return periods (year) atop Di Wang Tower were deter- Acceleration for amplitude-dependent damping ratio mined, as shown in Table 1. The wind-induced responses 0.6 of this building were evaluated when the mean wind speeds atop the building were 15 m/s, 27 m/s, 32 m/s, 37 m/s and 0.4

48 m/s, respectively, and the wind action was assumed to 0.2 be perpendicular to one side of the building. The mean wind speeds of 27 m/s, 32 m/s, 37 m/s and 48 m/s atop the Di 0 Acceleration (gal) Wang Tower are approximately corresponding to the -0.2 design wind speeds with 5-year, 10-year, 25-year and 100- year return periods, respectively. The time-step was selected -0.4 as 0.05 s and total time duration for the step-by-step -0.6 dynamic analysis was 10 min. In order to examine the time 0 20 40 60 80 100 120 140 160 180 domain computational results, the wind-induced responses Time (s) of this building were also calculated in frequency domain Fig. 7. Along-wind acceleration in X-direction atop the building (mean under the same load cases for comparison purposes. The wind speed = 15 m/s).

Table 1 Design wind speeds atop Di Wang Tower corresponding to diﬀerent return periods Return period (Year) 5 10 20 30 40 50 60 100 Wind speed (m/s) 27.72 31.96 36.19 38.50 40.42 41.58 42.74 45.80 J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176 1171

Table 2 Along-wind responses of Di Wang Tower in X-direction Wind speed Damping Results from the time domain analysis Results from the frequency domain (m/s) ratio analysis Displacement (mm) Acceleration (gal) RMS Maximum Peak RMS Maximum Peak RMS of RMS of value value factor value Value factor displacement (mm) acceleration (gal) 15 NL 2.32 6.62 2.85 0.24 0.70 2.91 1% 1.95 5.69 2.91 0.18 0.57 3.17 2.23 0.20 2% 1.70 4.75 2.79 0.12 0.38 3.17 2.01 0.14 3% 1.63 4.45 2.73 0.10 0.31 3.10 1.94 0.12 27 NL 6.72 19.32 2.88 0.73 2.39 3.27 1% 6.43 18.49 2.87 0.69 2.26 3.27 7.39 0.75 2% 5.34 15.01 2.81 0.49 1.61 3.29 6.14 0.59 3% 4.98 13.25 2.67 0.41 1.29 3.15 5.43 0.44 32 NL 11.04 29.50 2.67 1.22 3.31 2.71 1% 11.59 31.42 2.71 1.31 3.77 2.88 14.5 1.64 2% 9.04 25.17 2.78 0.91 2.53 2.78 11.38 1.16 3% 8.19 23.34 2.84 0.73 2.04 2.79 10.10 0.90 37 NL 15.52 39.79 2.56 1.31 3.55 2.71 1% 15.76 43.29 2.75 1.38 3.88 2.81 17.96 1.73 2% 15.13 38.60 2.55 1.21 3.46 2.86 16.80 1.28 3% 14.80 37.10 2.50 1.10 3.40 2.72 15.40 1.13 48 NL 24.11 68.29 2.83 2.39 6.57 2.74 1% 27.20 71.65 2.63 2.93 7.82 2.67 32.08 3.26 2% 23.72 68.91 2.90 2.31 6.71 2.90 26.73 2.58 3% 22.52 59.31 2.63 1.98 5.68 2.87 24.28 2.13 Note: NL – from the proposed amplitude-dependent damping model. mum responses of the along-wind responses of this tower time history acceleration and displacement responses at under other load cases are listed in Table 2. It is found that the top floor of the building are shown in Figs. 8 and 9 the results obtained from the frequency-domain analysis for the case of the mean wind speed atop the building as are generally slightly greater than those from the time 15 m/s. Comparing the results shown in Figs. 6 and 8,it domain method, which is considered to be reasonable seems that the difference between the results from the due to existence of time constant effects on the dynamic amplitude-dependent damping model and those from the responses, which will be discussed later. This implies that assumed constant damping ratio of 1% is relatively small. the time-domain analysis method developed in this study The RMS and maximum responses of the along-wind is reliable. The peak factor values determined from the responses of this building in Y-direction under other load time-domain analysis are normally between 2.7 and 3.3, cases are listed in Table 3. The peak factor values shown as shown in Table 2, which are smaller than the gener- in Table 3 are in the range of 2.5–3.6. Comparing these ally adopted value of 3.5 in wind engineering practices. results with those in Table 2, it is found that the along-wind The results listed in Table 2 clearly demonstrate the effect of amplitude-dependent damping on the along-wind responses of the building under different mean wind speeds. 1 Acceleration for constant damping ratio 1.0% Acceleration for amplitude-dependent damping ratio It is found that except several cases corresponding to 0.8 damping ratio f = 1%, the results for the constant damping 0.6 ratios are smaller than those predicted from the amplitude- 0.4 dependent damping model. This implies that the design 0.2 damping levels for the tall building, which were determined based on current design structural design codes; appear too 0

high and not conservative to estimate the wind-induced Acceleration (gal) -0.2 response of the building. -0.4

-0.6

5.2. The along-wind responses (in Y-direction) under -0.8 diﬀerent load cases 0 20 40 60 80 100 120 140 160 180 Time (s)

The along-wind responses in Y-direction were also Fig. 8. Along-wind acceleration in Y-direction atop the building (mean numerically estimated by the time domain method. The wind speed = 15 m/s). 1172 J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176

than 1% in X- and Y-direction when the mean wind speed 1 Acceleration for constant damping ratio 3.0% Acceleration for amplitude-dependent damping ratio atop the building is 27 m/s which is corresponding to the 0.8 design wind speed with 5-year return period, and their val- 0.6 ues are about 2% when the mean wind speed atop the 0.4 building is 48 m/s which is equivalent to the design wind

0.2 speed with 100-year return period. However, the damping ratios rarely reach 3% under all the load cases considered 0 in this study.

Acceleration (gal) -0.2

-0.4 5.3. Evaluation of the across-wind response (in X-direction) -0.6

-0.8 When the building vibrates in along-wind direction (e.g., 0 20 40 60 80 100 120 140 160 180 Time (s) in Y-direction), its across-wind vibration also occurs simul- taneously (in X-direction). In this section, the time history Fig. 9. Along-wind acceleration in Y-direction atop the building (mean across-wind responses in X-direction were calculated by the wind speed = 15 m/s). similar method as that described in Section 5.2. The time history acceleration and displacement responses at the responses in Y-direction are much greater than those in X- top ﬂoor of the building are shown in Figs. 10 and 11 for direction. This is due to the fact that the lateral stiﬀness of the case of the mean wind speed atop the building as the building in Y-direction is relatively weak so that it is 27 m/s. The RMS and maximum values of the across-wind more sensitive to wind excitation. response in X-direction under other load cases are listed in Wind-induced response of a tall building is usually dom- Table 4. It can be seen from this table that the peak factor inated by its fundamental vibration mode. The full-scale is also generally between 2.7 and 3.6. Comparing the results measurements indicated that the actual damping ratios of with those presented in Tables 2 and 3, it is found that the the fundamental vibration mode of this building in X-direc- across-wind responses in X-direction are greater than the tion and Y-direction all increase with the mean wind speed along-wind responses in X-direction and are smaller than and the wind-induced responses. Comparison of the the along-wind responses in Y-direction for the same responses presented in Tables 2 and 3 reveals that the wind speed actions. This clearly demonstrates that the actual damping ratios of the fundamental modes are less across-wind responses are comparable to the along-wind

Table 3 Along-wind responses of Di Wang Tower in Y-direction Wind speed Damping Results from the time domain analysis Results from the frequency domain (m/s) ratio Displacement (mm) Acceleration (gal) analysis RMS Maximum Peak RMS Maximum Peak RMS of RMS of value value factor value value factor displacement (mm) acceleration (gal) 15 NL 4.65 13.26 2.85 0.38 1.17 3.07 1% 4.28 13.20 3.08 0.31 0.97 3.12 4.85 0.39 2% 3.97 13.00 3.27 0.24 0.75 3.13 4.08 0.28 3% 3.79 12.29 3.24 0.20 0.66 3.30 3.89 0.22 27 NL 25.17 67.58 2.68 2.49 6.83 2.74 1% 24.18 66.90 2.76 2.34 6.15 2.62 25.30 2.01 2% 18.57 48.89 2.63 1.61 4.57 2.83 20.62 1.41 3% 16.12 41.49 2.57 1.26 3.68 2.92 18.78 1.15 32 NL 30.38 93.33 3.07 2.76 9.30 3.36 1% 31.00 95.59 3.08 2.76 9.24 3.34 38.90 3.19 2% 26.74 85.17 3.18 2.06 7.32 3.55 31.10 2.25 3% 24.61 77.69 3.16 1.70 6.18 3.63 28.06 1.83 37 NL 41.10 104.40 2.54 3.40 9.70 2.85 1% 41.72 107.40 2.57 3.32 9.40 2.81 45.83 3.62 2% 39.20 99.09 2.53 2.77 6.97 2.51 38.80 3.06 3% 36.40 89.70 2.46 2.27 5.85 2.57 35.5 2.64 48 NL 67.41 197.30 2.93 6.64 22.05 3.32 1% 85.18 248.14 2.91 8.60 28.25 3.29 97.9 9.57 2% 65.49 195.94 2.99 6.02 19.38 3.22 72.7 6.77 3% 56.72 170.10 3.01 4.81 15.29 3.19 62.83 5.51 Note: NL – from the proposed amplitude-dependent damping model. J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176 1173

2.5 2.5 Acceleration for constant damping ratio 1.0% Acceleration for constant damping ratio 3.0% 2 Acceleration for amplitude-dependent damping ratio 2 Acceleration for amplitude-dependent damping ratio

1.5 1.5

1 1

0.5 0.5

0 0

-0.5 -0.5 Acceleration (gal) Acceleration (gal) -1 -1

-1.5 -1.5

-2 -2

0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 Time (s) Time (s)

Fig. 10. Across-wind acceleration in X-direction atop the building (mean Fig. 11. Across-wind acceleration in X-direction atop the building (mean wind speed = 27 m/s). wind speed = 27 m/s). responses even if the super tall building is subject to mod- are relatively smaller than those from the frequency erate wind speed excitation. The effect of actual amplitude- domain method. The main reason for the difference may dependent damping on the across-wind responses at be due to the effect of time constant on the dynamic different wind speed levels was clearly demonstrated from response. As indicated by Jeary [10], a structure with a res- the calculated results in Table 4. onance frequency of f0 and an associated damping ratio n0 has a time constant s which is defined as follows: 6. Effect of time constant on the along-wind responses of the 1 tall building s ¼ ð15Þ 2pf0n0 As indicated in Tables 2 and 3, the results of the wind- When the along-wind response of a tall building is evalu- induced responses obtained from the time domain analysis ated in frequency domain, the ratio of the RMS deflection

Table 4 The across-wind responses of Di Wang Tower in X-direction Wind speed Damping Results from the time domain method Results from the frequency domain (m/s) ratio Displacement (mm) Acceleration (gal) method RMS Maximum Peak RMS Maximum Peak RMS for RMS for value value factor value value factor displacement (mm) acceleration (gal) 15 NL 2.93 8.20 2.79 0.26 0.81 3.10 1% 2.86 8.03 2.81 0.24 0.84 3.50 2.90 0.23 2% 2.78 7.50 2.66 0.19 0.69 3.56 2.70 0.16 3% 2.66 7.05 2.68 0.16 0.58 3.60 2.58 0.13 27 NL 13.60 34.90 2.56 0.82 2.71 3.30 1% 14.10 36.60 2.59 1.20 3.12 2.60 14.47 1.30 2% 12.65 33.83 2.59 0.77 2.51 3.26 13.02 0.91 3% 11.65 31.79 2.54 0.65 2.40 3.70 12.09 0.74 32 NL 22.10 62.24 2.81 1.36 4.86 3.57 1% 23.19 68.56 2.95 1.69 5.16 3.05 24.92 2.03 2% 21.94 62.79 2.86 1.32 4.79 3.63 22.60 1.50 3% 20.41 60.13 2.95 1.16 4.75 4.09 21.30 1.22 37 NL 33.54 97.26 2.91 2.19 8.04 3.67 1% 36.38 100.68 2.76 2.74 9.46 3.45 37.05 3.26 2% 29.83 83.82 2.83 2.09 7.94 3.79 32.19 2.29 3% 26.91 78.85 2.91 1.74 6.11 3.51 28.13 1.87 48 NL 65.10 187.04 2.87 4.63 14.09 3.03 1% 75.74 208.94 2.75 6.82 17.55 2.57 79.69 6.93 2% 67.21 189.02 2.81 5.14 15.37 2.99 68.35 5.38 3% 63.42 188.69 2.97 4.16 12.86 3.09 63.03 4.27 Note: NL – proposed amplitude-dependent damping model. 1174 J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176 rx to the mean deflection x atop the building can be ob- ture is modeled as a multi-degree of freedom system in the tained by the following equation [7]: dynamic analysis. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Based on the generated along-wind fluctuating forces, rx SE the along-wind responses in Y-direction of the building ¼ 2Iu B þ ð16Þ x n0 were determined in both time and frequency domain for the case that the mean wind speed atop the building was 32 m/s which is corresponding to the design wind speed where Iu is the turbulence intensity atop the building, B is the background excitation factor, S is the size reduction with 10-year return period. For various simulation config- factor to account for the correlation of pressure over the urations, it was assumed that except for the damping ratios building, E is the gust wind energy factor which is defined of the building, the structural parameters and its dynamics p characteristics were kept as unchanged for examining the by the equation E ¼ 4 f0Sðf0Þ, where S(f0) is the value of normalized fluctuating along-wind wind speed spectrum effect of time constant on the wind-induced responses of Di Wang Tower. While the damping ratios were supposed at natural frequency f0 and it can be calculated from Eq. (6). to take 13 different constant values (0.01–5% as indicated The acceleration response can be estimated by Eq. (17) in Table 5), the time constant varied in the range of in the way that the background displacement response con- 18.59–930 s/rad. The ratios of RMS responses obtained tributes almost nothing to the RMS acceleration: by the time domain analysis to those by the frequency sffiffiffiffiffiffi domain method actually reflect the effect of the time con- rx 2 SE stant on the along-wind responses of the tall building. ¼ð2pf0Þ 2Iu ð17Þ These ratios are also shown in Fig. 12. It can be seen from x n0 this figure that there is a systematic decrease in the ratio of the simulated time domain results to the ‘‘expected’’ fre- The relationship established in Eqs. (16) and (17) assumes that no effect caused by systematic response is presented [10]. In practice, a systematic effect actually exists for a damped structural system. For a linear damped system 1 responding to a specified load with single frequency sine Reduction factor for acceleration 0.95 wave, it will reach 99.3% of its final steady-state response Reduction factor for displacement only after 5 time constants (5s) [10]. The structure may 0.9 never reach the response estimated from Eqs. (16) and (17) because of the lag between the actual response and 0.85 the ‘‘final’’ steady-state response. It was found that the 0.8 reduction in RMS displacement of a structure with a long time constant can be as high as 20% [10]. This value is cor- 0.75 responding to a reduction factor of 80%. By taking Di 0.7 Wang Tower as an example, the effect of time constant 0 100 200 300 400 500 600 700 800 900 1000 on its wind-induced responses will be studied herein. Un- Ratio of time-dimain to frequency domain value Time constant (sec/rad) like the previous study on this topic [10], tall building struc- Fig. 12. The effect of time constant on the along-wind responses.

Table 5 Along-wind responses in Y-direction (mean wind speed is 32 m/s atop the building) Damping Time constant Results from the frequency domain Results from the time Ratio of the results from the ratio (%) (s/rad) analysis domain analysis time domain to those from the frequency domain RMS of RMS of RMS of RMS of Acceleration Displacement acceleration (gal) displacement (mm) acceleration (gal) displacement (mm) 0.1 930.1 10.29 108.5 7.59 79.21 0.739 0.730 0.2 465.1 7.04 75.8 5.39 57.46 0.766 0.754 0.4 232.5 5.05 56.43 4.13 44.03 0.819 0.781 0.8 116.7 3.51 41.80 3.10 33.85 0.885 0.806 1 92.9 3.19 38.90 2.76 31.00 0.869 0.810 1.5 61.9 2.60 33.90 2.34 28.56 0.884 0.847 2 46.5 2.25 31.10 2.06 26.24 0.917 0.862 2.5 37.1 2.01 29.30 1.84 25.30 0.917 0.862 3 31.1 1.83 28.06 1.70 24.61 0.934 0.877 3.5 24.3 1.69 27.10 1.57 23.85 0.934 0.884 4 23.3 1.58 26.30 1.48 23.32 0.943 0.893 4.5 20.6 1.49 25.78 1.41 23.04 0.952 0.847 5 18.7 1.41 25.30 1.35 22.79 0.962 0.901 J.R. Wu et al. / Computers and Structures 85 (2007) 1165–1176 1175 quency domain steady responses as the time constant to be about 2% when the mean wind speed atop the increases. The maximum of the reduction factor can reach building was 48 m/s which is approximately equiva- 73% over the range of the time constants considered in this lent to the design wind speed with 100-year return study. As modern tall buildings are becoming taller and rel- period. This suggests that damping ratio of 1.0–2% atively more flexible, there is a trend for the natural fre- critical appears reasonable for wind-resistant design quency to become lower and damping ratios to become of the super tall building. smaller. Therefore, the time constants of these kinds of (4) The effect of the time constant on the wind-induced modern tall buildings may be greater than those of most responses of Di Wang Tower is becoming more obvi- buildings built in the past, and the reduction of the wind- ous when the value of the time constant reaches a cer- induced response by time domain method results in a tain value. There is a systematic decrease in the ratio benefit for reducing design wind loads on such high-rise of time domain result to the ‘‘expected’’ steady buildings (Fig. 12). response as the time constant increases. The maximum of reduction factor can be as high as 73% over 7. Conclusion the range of time constants investigated in this study, suggesting that the adoption of the time domain Based on long-term full-scale measurements of wind- method offers a benefit for reducing design wind induced responses of Di Wang Tower under typhoon condi- loads on the super tall building. tions, an empirical amplitude-dependent damping model for this building was proposed in this study. The damping data- Acknowledgements base was established during the passages of several typhoons including two very strong typhoons. The wind-induced The work described in this paper was fully supported by responses of the super tall building were numerically evalu- a grant from City University of Hong Kong (Project No. ated in time domain and also in frequency domain. The 7001972). The financial support is gratefully acknowl- amplitude-dependent damping effects on the wind-induced edged. Thanks are due to Prof. A.P. Jeary, Dr. N. 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