# Analytical Simulation of 3D Wind-Induced Vibrations of Rectangular Tall Buildings in Time Domain

Authors: Iván F Huergo; Hugo Hernández-Barrios; Roberto Gómez-Martíne
Journal: Shock and Vibration
 (2022)

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## Abstract

The Monte Carlo simulation method for along-wind loads on tall buildings performed in the time and space domain may be the only analytical viable option for specific problems such as nonlinearity behavior, nonclassically damping, and detailed structural models in commercial software. However, both across-wind and torsional-wind loads due to vortex shedding are not usually simulated in time domain because the vertical decay constants are unknown or the empirical coherence functions cannot be applied in Monte Carlo simulation methods. In this paper, the spectral representation (SR) method is used to simulate in time domain the along-wind, across-wind, and torsional-wind loads on rectangular tall buildings considering the vertical correlation between the signals. The Krenk root-coherence function is used for the normalized cross-spectrum on the along-wind direction, whereas the Davenport root-coherence function is used for the other two types of wind loads. For both across-wind and torsional-wind loads, the Davenport root-coherence function was assessed at the vortex shedding frequency by changing the vertical decay constants until converge between the Davenport model and the Liang empirical coherence model was achieved. Based on a three‐dimensional model with two translational and one torsional degree of freedom for each floor, the proposed vertical decay constants were validated by comparing the elastic response between frequency domain and time domain approaches. Generally speaking, the results show that peak displacements are significantly underestimated for both across-wind and torsional directions when vertical correlation is neglected. In addition, the advantages of time domain simulation were shown by performing a nonlinear time history analysis considering a bilinear isotropic material hardening model in both translational directions.


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## Body

## 1. Introduction

Tall buildings are usually susceptible to wind-induced vibrations, which are usually computed analytically in the frequency domain by using empirical cross-spectrum density functions of wind loads and simplified continuous beam models [1, 2]. However, some particular problems hinder the use of a frequency domain approach to compute the wind-induced response such as nonlinearity behavior (hysteric loops), nonclassical damping (soil-structure interaction and passive energy dissipation devices) [3–6], and detailed structural models in commercial software where the user can only define time history functions as dynamic loads. Therefore, the temporal response analysis has to be performed in such cases rather than the analysis in the frequency domain.The autoregressive moving average (ARMA) algorithm [7] and the spectral representation (SR) model [8] are the most commonly used methods to simulate along-wind loads on tall buildings in time domain. The essential feature of the SR method [8] is that a random process can be simulated by a series of cosine functions with random frequency; then, the density function of the random frequency is derived from the specified cross-spectral density matrix for multivariate process or from the specified spectral density function for a multidimensional process. It is worth mentioning that the summation of a large set of trigonometric terms involved in the SR method renders the approach computational less efficient. On the other hand, the parameters associated with ARMA models are determined in such a manner that the spectral description of the system response to white noise approximates the target spectral characteristics in an optimum sense. However, for the multivariate ARMA models such as the wind field simulation, a large number of nonlinear equations must be solved, which is why the SR method is usually preferred.For along-wind loads, the SR method is based on the assumption that fluctuating components of the wind velocity field can be idealized as a mean-zero multivariate and multidimensional Gaussian process, where the normalized cross-spectrum describes the statistical dependence between the turbulent components at two points at a given frequency. The real part of the normalized cross-spectrum is called the normalized cospectrum, and the root-coherence function is defined as the absolute value of the normalized cross-spectrum. On a purely empirical basis, Davenport [9] originally suggested an exponential expression as the root-coherence function, where the vertical decay constant controls the vertical correlation between turbulence at two points. The vertical decay constant usually ranges from 7 to 11.5 [9–12].The empirical root-coherence function proposed by Davenport [9] has the advantage of simplicity, but incorporates two inconsistencies [11]: (1) the function is positive for any separation, which is in conflict with the definition of the longitudinal turbulence component with a zero mean; (2) the normalized cospectrum approaches unity for small frequencies, which is not true for separations of the same order of magnitude or even larger than the average size of gusts, where the wind structure is characterized by a lack of correlation even at low frequencies. Consequently, Krenk [13] derived a simple modified exponential format not encumbered by the two inconsistencies mentioned above. Using full-scale measurements with vertical separations of the order of 10–20 m, Hansen and Krenk [14] determined that a vertical decay constant of 5 corresponds to a vertical decay constant of 7.5 in the exponential format proposed by Davenport [9], i.e., close to the vertical decay constant of 8 used by some building codes. Different power-spectral density (PSD) functions have been successfully used to simulate the turbulent along-wind velocity component [15–18]. However, these wind time series are usually simulated using the exponential format [9] instead of the modified exponential format [13]. It is worth mentioning that there are other models for the root-coherence function of the along-wind turbulent component, such as the Frøya model [19] and the IEC model [20]; however, both models are usually used to simulate along-wind loads on offshore wind turbines.Based on the above, different kinds of power-spectral density (PSD) functions can be used to simulate along-wind loads of line-like structures by using the SR method. However, Bojórquez et al. [21] demonstrated that the PSD functions of Von Kármán, Von Kármán Harris, and Solari generated wind time series with better characteristic of turbulence intensity and scale length, compared with the models of Davenport, Kaimal, and modified Kaimal.For tall buildings with aspect ratios of over 3, their across-wind responses usually exceed along-wind responses and can even reach them several times. Across-wind dynamic load on tall buildings is induced by three mechanisms: (1) along-wind turbulence, (2) across-wind turbulence, and (3) wake excitation (vortex shedding). Vortex shedding is the main contributor to the across-wind response of a tall building, particularly when either of its lowest translational natural frequencies approaches the vortex shedding frequency. In addition to transmission lines, bridge pylons, and bridge deck sections, potential super slim and tall buildings can also be subjected to self-excited forces in the across-wind direction; that is, galloping. In fact, the galloping effect can be measured in a wind tunnel by using different methodologies like the hybrid aeroelastic-pressure balance technique [22].Generally speaking, wind-induced vibrations can be reduced in three different ways: (1) by changing the stiffness or mass, (2) by increasing the damping with passive or active control devices [3–6, 23–26], and (3) by reducing wind loads with aerodynamic shapes such as tapering. The fact that a tapered tall building might spread the vortex shedding over a broad range of frequencies makes it more effective for reducing across-wind responses has been established [27]. However, Chen et al. [28] discovered a new phenomenon called partial reattachment, which suppresses vortex shedding in reattached regions and forms a separation envelope. Based on the partial reattachment phenomenon, tapering promotes the unsteady effect, and galloping interaction near the free end, making the prism more susceptible to wind-induced vibrations. Overall, tapering promotes a structure’s susceptibility to wind-induced vibrations and may cause unanticipated perils in engineering practice [28].For tapering cylinder-like structures, Vickery and Clark [29] proposed a formula for the normalized across-wind load, whereas Gu and Quan [30] proposed new formulas for the power spectra of the base moment induced by across-wind loads for 15 typical tall building models. However, these power spectra are not useful to simulate across-wind loads along the height of the building. In fact, the literature features some empirical formulas for the power spectra of across-wind loads on square cross section buildings [31, 32] and rectangular tall buildings [1, 33] as a result of wind tunnel tests.Tsukagoshi et al. [15] simulated across-wind loads on a square tall building using the power spectrum proposed by Ohkuma and Kanaya [31], the empirical root-coherence function proposed by Davenport [9], and a vertical decay constant equal to 3. However, Tsukagoshi et al. [15] did not justify the value of the vertical decay constant used in the simulation process. On the other hand, Slooten [34] simulated across-wind loads on a rectangular tall building by using the power spectrum proposed by Liang et al. [1]; however, Slooten [34] did not consider the vertical correlation between the signals, which is physically wrong.Wind-induced torsional vibration of tall buildings can enlarge the dynamic response near the peripheries of their cross section, especially when the side faces of a rectangular tall building are wider, and/or it is asymmetric, and/or its lowest torsional natural frequency approaches either of its lowest translational natural frequencies. Choi [35] and Carini [36] studied the wind-induced torsional response on tall buildings using experimental data, whereas Liang et al. [2] proposed an empirical formula for the power spectra of torsional-wind loads on rectangular tall buildings.The most widely used empirical formulas for across-wind force spectra and torque spectra are those proposed by Liang et al. [1, 2], which are valid for rectangular buildings with various side ratios at normal attack angles. Based on the experimental research, Liang et al. [1, 2] also proposed empirical formulas for the root-coherence function of across-wind loads and torque between two different levels within the spectral peaks. However, these empirical formulas of vertical correlation cannot be used directly by the SR method because they do not depend on the frequency content of the spectra and lead to a negative definite coherence matrix that cannot be factorized through the Cholesky decomposition.In this paper, a review of the parameters involved in simulating wind-induced loads on rectangular tall buildings was carried out in order to unify the simulation process by using the SR method. The vertical decay constants for both across-wind and torsional-wind loads are proposed based on a comparative analysis between Liang coherence function [1, 2] and Davenport coherence function [9], In this way, both correlated and uncorrelated simulated signals are used in time domain dynamic analysis in order to study the effect of vertical correlation on 3D wind-induced vibrations of rectangular tall buildings, particularly for across-wind and torsional responses. Furthermore, a nonlinear time history analysis was performed to demonstrate the advantages of performing a time domain simulation against the frequency domain approach.


## 2. Wind Loads on Tall Buildings

Under the action of turbulent wind, tall buildings are loaded simultaneously in the along-wind, across-wind, and torsional directions as shown in Figure1. When a vortex forms on the side of a building, it creates a suction force that can induce large amplitude vibrations in the plane normal to the wind when the vortex shedding is in resonance with one of the natural frequencies of vibration of the building. In a similar way, torque is mainly induced by the asymmetric distribution of wind pressure caused by vortex shedding.Figure 1
Wind-induced responses of tall buildings.Based on Figure1, wind loads on a uniform N-story building are given by(1)FDzj,t=12ρaAjCDU¯zj+uzj,t2,FLzj,t=12ρaAjF¯Lzj,tU¯zj2,Mzj,t=12ρaAjBM¯zj,tU¯zj2,where j = 1,2,3, …, N; FDzj,t is the along-wind force at height zj at time t; FLzj,t is the across-wind force at height zj at time t; Mzj,t is the wind-induced torque at height zj at time t; ρa is the density of air, which depends on the barometric pressure and the average annual temperature; pw is the windward pressure; pl is the leeward pressure; Uzj,t is the wind velocity in the longitudinal direction at height zj at time t; U¯zj is the mean wind velocity at height zj, that is, the mean value of Uzj,t; uzj,t is the turbulent component in the wind direction at height zj at time t, that is, the fluctuating component of Uzj,t; Aj is the projected area of the jth story, which is taken perpendicular to Uzj,t; B is the width of the windward side; D is the depth of the cross section of the building; H is the total height of the building; CD is the drag coefficient (see Appendix A); F¯Lzj,t is the nondimensional lift force at height zj at time t; and M¯zj,t is the nondimensional torque at height zj at time t.


### 2.1. Mean Wind Velocity

In practical wind load codes and standards, the power law and the logarithmic law are used to describe the wind profile in the atmospheric boundary layer. However, neither law is valid at very high altitudes above ground [11]. A more precise expression based on the mathematical model developed by Harris and Deaves [37] is the corrected logarithmic law, which fits the experimental data accurately and covers surface roughness changes. According to Harris and Deaves [37], the mean wind velocity is given by(2)U¯z=u∗κlnzzo+5.75zzg−1.88zzg2−1.33zzg3+0.25zzg4,where(3)u∗=κU¯zrlnzr/zo,zg=u∗6fc,fc=2Ωsinλ,u∗ is the friction velocity, κ is the Von Kármán´s constant (κ∼ 0.4), zo is the surface roughness length, zg is the gradient height, zr is the reference height (zr=10m), fc is the Coriolis parameter, Ω is the angular velocity of the Earth (Ω=7.27×10−5rad/s), and λ is the latitude. According to Simiu and Scanlan [38], typical values of zo for various types of built-up terrain are shown in Table 1.Table 1
Values of surface roughness length for various types of built-up terrain.
Type of surfacezo(m)Sand0.0001–0.001Snow surface0.001–0.006Mown grass0.001–0.01Low grass, steppe0.01–0.04Fallow field0.02–0.03High grass0.04–0.1Palmetto0.1–0.3Pine forest0.9–1Sparsely built-up suburbs0.2–0.4Densely built-up suburbs, towns0.8–1.2Centers of large cities2–3


### 2.2. Power Spectral Density Functions

The power-spectral density (PSD) function of a time series describes the distribution of power into the frequency components composing that signal, that is, the variations (energy) as a function of frequency. The unit of a PSD function is energy (variance) per frequency (width), and the energy within a specific frequency range can be obtained by integrating the PSD function within that frequency range. Generally speaking, the PSD function of a stationary stochastic processgjt can be defined as(4)Sgzj,f=Rgzj,fσg2zjf,where the subscript g is referred to the stationary stochastic process gjt, that is, uzj,t, FLzj,t,or Mzj,t; Rgzj,f is the nondimensional PSD function of gjt; and σgzj is the standard deviation of gjt. For rectangular tall buildings, the PSD functions of uzj,t, FLzj,t, and Mzj,t are shown in Appendices A, B, and C, respectively.


### 2.3. Vertical Correlation between Two Points

Tall buildings are usually modeled as line-like structures, which means that wind loads are simulated along thez-axis. The statistical dependence between the power spectra at two points Pj0,0,zj and Pk0,0,zk at frequency f is given by [11](5)Cohjkf=SjkfSgzj,fSgzj+rz,f,where Cohjkf is the root-coherence function between the two points; Sjkf is the cross-spectrum of the two signals at points Pj0,0,zj and Pk0,0,zk, respectively; rz is the vertical separation between the two points; Sgzj,f is the PSD function of the stochastic process at point Pj0,0,zj, and Skzj+rz,f is the PSD function of the stochastic process at point Pk0,0,zk.


### 2.4. Strouhal Number

In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff body at certain velocities, depending on the size and shape of the body. In this flow, vortices are created at the back of the body and detach periodically from either side of the body forming a Von Kármán vortex street. The fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object (see Figure1). The object will tend to move toward the low-pressure zone. If the frequency of vortex shedding matches the resonance frequency of the structure, then the structure may begin to resonate in the across-wind and torsional-wind directions. Accordingly, the vortex shedding frequency is given by(6)fsz=StU¯zB,where St is the Strouhal number, which is a nondimensional quantity describing oscillating flow mechanisms. When 1/4≤D/B≤4, the Strouhal number for across-wind loads on rectangular tall buildings is given by [1](7)St=0.0941/4≤D/B≤1/2,0.002D/B2−0.023D/B+0.1051/2<D/B≤4,and the Strouhal number for wind-induced torque is given by [2](8)St=0.0811/4≤D/B≤1/2,0.032D/B−0.12D/B+0.151/2<D/B≤4.


### 2.5. Stochastic Simulation by Using the Spectral Representation Method

In this paper, the bandwidth of the power-spectral density (PSD) function is ranged fromf1=0.001Hz to fm=10Hz in order to include the wind gust frequencies and the structural frequencies of any tall building. Based on the Nyquist theorem, the sampling interval for the signals is given by(9)Δt=12fm,and the sampling frequency for the PSD function is given by(10)Δf=Δtfm−f1Ts,where Ts is the total duration of the signals, which is usually equal to 600 s.Let us consider a set ofN homogeneous Gaussian multidimensional processes gjt(j = 1,2,3, …, N) with mean-zero and with the cross-spectral density matrix Sggfn defined by(11)Sggfn=S11fnS12fn⋯S1NfnS21fnS22fn⋯S2Nfn⋮⋮⋱⋮SN1fnSN2fn⋯SNNfn,where(12)Sjkfn=CohjkfnSgzj,fnSgzk,fn,Cohjkfn is the root-coherence function between the two points at frequency fn, Sgzj,fn is the PSD function at point Pj0,0,zj at frequency fn , Sgzk,fn is the PSD function at point Pk0,0,zk at frequency fn, and n=1,2,3,...,m. Through the Cholesky decomposition, the cross-spectral density matrix can be decomposed into the following format:(13)Sggfn=HfnHfnT,where(14)Hfn=H11fn0⋯0H21fnH22fn⋯0⋮⋮⋱⋮HN1fnHN2fn⋯HNNfn,and the superscript T represents matrix transpose.The power spectra of both the across-wind loads and wind-induced torque may lead to a negative definite Hermitian cross-spectral density matrix that cannot be decomposed through the Cholesky factorization. The SR method, based on the Cholesky decomposition of the lagged coherency matrix [39], is another form of the SR method that can solve this problem if certain types of empirical root-coherence functions are used. Alternatively, the cross-spectral density matrix can be expressed as follows [39]:(15)Sggfn=ΨfnΓfnΨfnT,where(16)Ψfn=Sgz1,fnSgz2,fn⋱SgzN,fn,Γfn=1Coh12fn…Coh1NfnCoh21fn1…Coh2Nfn⋮⋮⋱⋮CohN1fnCohN2fn⋯1.For certain types of empirical root-coherence functions, the coherence matrixΓfn is a nonnegative definite Hermitian matrix and can be factorized through the Cholesky decomposition(17)Γfn=EfnEfnT,where Efn is a lower triangular matrix expressed as(18)Efn=ε11fn0⋯0ε21fnε22fn⋯0⋮⋮⋱⋮εN1fnεN2fn⋯εNNfn,then, the stationary stochastic processgjt(j = 1,2,3, …, N) can be simulated by the following equation (8):(19)gjt=∑k=1N∑n=1mHjkfn2Δfcos2πfnt+ϕkn,where(20)Hjkfn=εjkfnSgzj,fn,and ϕkn are independent random phase angles uniformly distributed between 0 and 2π.Based on (19), the longitudinal turbulent component of wind velocity uzj,t, the across-wind load FLzj,t , and the wind-induced torque Mzj,t, can be simulated with prior knowledge of the PSD functions and the empirical root-coherence functions. For the along-wind direction, the simulated turbulent component at height zj must be filtered by multiplying the aerodynamic admittance function (see Appendix A) by the discrete Fourier transform of uzj,t; subsequently, uzj,t must be retrieved into the time domain through the inverse discrete Fourier transform in order to compute the along-wind load using Equation (1).


## 2.1. Mean Wind Velocity

In practical wind load codes and standards, the power law and the logarithmic law are used to describe the wind profile in the atmospheric boundary layer. However, neither law is valid at very high altitudes above ground [11]. A more precise expression based on the mathematical model developed by Harris and Deaves [37] is the corrected logarithmic law, which fits the experimental data accurately and covers surface roughness changes. According to Harris and Deaves [37], the mean wind velocity is given by(2)U¯z=u∗κlnzzo+5.75zzg−1.88zzg2−1.33zzg3+0.25zzg4,where(3)u∗=κU¯zrlnzr/zo,zg=u∗6fc,fc=2Ωsinλ,u∗ is the friction velocity, κ is the Von Kármán´s constant (κ∼ 0.4), zo is the surface roughness length, zg is the gradient height, zr is the reference height (zr=10m), fc is the Coriolis parameter, Ω is the angular velocity of the Earth (Ω=7.27×10−5rad/s), and λ is the latitude. According to Simiu and Scanlan [38], typical values of zo for various types of built-up terrain are shown in Table 1.Table 1
Values of surface roughness length for various types of built-up terrain.
Type of surfacezo(m)Sand0.0001–0.001Snow surface0.001–0.006Mown grass0.001–0.01Low grass, steppe0.01–0.04Fallow field0.02–0.03High grass0.04–0.1Palmetto0.1–0.3Pine forest0.9–1Sparsely built-up suburbs0.2–0.4Densely built-up suburbs, towns0.8–1.2Centers of large cities2–3


## 2.2. Power Spectral Density Functions

The power-spectral density (PSD) function of a time series describes the distribution of power into the frequency components composing that signal, that is, the variations (energy) as a function of frequency. The unit of a PSD function is energy (variance) per frequency (width), and the energy within a specific frequency range can be obtained by integrating the PSD function within that frequency range. Generally speaking, the PSD function of a stationary stochastic processgjt can be defined as(4)Sgzj,f=Rgzj,fσg2zjf,where the subscript g is referred to the stationary stochastic process gjt, that is, uzj,t, FLzj,t,or Mzj,t; Rgzj,f is the nondimensional PSD function of gjt; and σgzj is the standard deviation of gjt. For rectangular tall buildings, the PSD functions of uzj,t, FLzj,t, and Mzj,t are shown in Appendices A, B, and C, respectively.


## 2.3. Vertical Correlation between Two Points

Tall buildings are usually modeled as line-like structures, which means that wind loads are simulated along thez-axis. The statistical dependence between the power spectra at two points Pj0,0,zj and Pk0,0,zk at frequency f is given by [11](5)Cohjkf=SjkfSgzj,fSgzj+rz,f,where Cohjkf is the root-coherence function between the two points; Sjkf is the cross-spectrum of the two signals at points Pj0,0,zj and Pk0,0,zk, respectively; rz is the vertical separation between the two points; Sgzj,f is the PSD function of the stochastic process at point Pj0,0,zj, and Skzj+rz,f is the PSD function of the stochastic process at point Pk0,0,zk.


## 2.4. Strouhal Number

In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff body at certain velocities, depending on the size and shape of the body. In this flow, vortices are created at the back of the body and detach periodically from either side of the body forming a Von Kármán vortex street. The fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object (see Figure1). The object will tend to move toward the low-pressure zone. If the frequency of vortex shedding matches the resonance frequency of the structure, then the structure may begin to resonate in the across-wind and torsional-wind directions. Accordingly, the vortex shedding frequency is given by(6)fsz=StU¯zB,where St is the Strouhal number, which is a nondimensional quantity describing oscillating flow mechanisms. When 1/4≤D/B≤4, the Strouhal number for across-wind loads on rectangular tall buildings is given by [1](7)St=0.0941/4≤D/B≤1/2,0.002D/B2−0.023D/B+0.1051/2<D/B≤4,and the Strouhal number for wind-induced torque is given by [2](8)St=0.0811/4≤D/B≤1/2,0.032D/B−0.12D/B+0.151/2<D/B≤4.


## 2.5. Stochastic Simulation by Using the Spectral Representation Method

In this paper, the bandwidth of the power-spectral density (PSD) function is ranged fromf1=0.001Hz to fm=10Hz in order to include the wind gust frequencies and the structural frequencies of any tall building. Based on the Nyquist theorem, the sampling interval for the signals is given by(9)Δt=12fm,and the sampling frequency for the PSD function is given by(10)Δf=Δtfm−f1Ts,where Ts is the total duration of the signals, which is usually equal to 600 s.Let us consider a set ofN homogeneous Gaussian multidimensional processes gjt(j = 1,2,3, …, N) with mean-zero and with the cross-spectral density matrix Sggfn defined by(11)Sggfn=S11fnS12fn⋯S1NfnS21fnS22fn⋯S2Nfn⋮⋮⋱⋮SN1fnSN2fn⋯SNNfn,where(12)Sjkfn=CohjkfnSgzj,fnSgzk,fn,Cohjkfn is the root-coherence function between the two points at frequency fn, Sgzj,fn is the PSD function at point Pj0,0,zj at frequency fn , Sgzk,fn is the PSD function at point Pk0,0,zk at frequency fn, and n=1,2,3,...,m. Through the Cholesky decomposition, the cross-spectral density matrix can be decomposed into the following format:(13)Sggfn=HfnHfnT,where(14)Hfn=H11fn0⋯0H21fnH22fn⋯0⋮⋮⋱⋮HN1fnHN2fn⋯HNNfn,and the superscript T represents matrix transpose.The power spectra of both the across-wind loads and wind-induced torque may lead to a negative definite Hermitian cross-spectral density matrix that cannot be decomposed through the Cholesky factorization. The SR method, based on the Cholesky decomposition of the lagged coherency matrix [39], is another form of the SR method that can solve this problem if certain types of empirical root-coherence functions are used. Alternatively, the cross-spectral density matrix can be expressed as follows [39]:(15)Sggfn=ΨfnΓfnΨfnT,where(16)Ψfn=Sgz1,fnSgz2,fn⋱SgzN,fn,Γfn=1Coh12fn…Coh1NfnCoh21fn1…Coh2Nfn⋮⋮⋱⋮CohN1fnCohN2fn⋯1.For certain types of empirical root-coherence functions, the coherence matrixΓfn is a nonnegative definite Hermitian matrix and can be factorized through the Cholesky decomposition(17)Γfn=EfnEfnT,where Efn is a lower triangular matrix expressed as(18)Efn=ε11fn0⋯0ε21fnε22fn⋯0⋮⋮⋱⋮εN1fnεN2fn⋯εNNfn,then, the stationary stochastic processgjt(j = 1,2,3, …, N) can be simulated by the following equation (8):(19)gjt=∑k=1N∑n=1mHjkfn2Δfcos2πfnt+ϕkn,where(20)Hjkfn=εjkfnSgzj,fn,and ϕkn are independent random phase angles uniformly distributed between 0 and 2π.Based on (19), the longitudinal turbulent component of wind velocity uzj,t, the across-wind load FLzj,t , and the wind-induced torque Mzj,t, can be simulated with prior knowledge of the PSD functions and the empirical root-coherence functions. For the along-wind direction, the simulated turbulent component at height zj must be filtered by multiplying the aerodynamic admittance function (see Appendix A) by the discrete Fourier transform of uzj,t; subsequently, uzj,t must be retrieved into the time domain through the inverse discrete Fourier transform in order to compute the along-wind load using Equation (1).


## 3. Vertical Decay Constants

### 3.1. Along-Wind Loads

On a purely empirical basis, Davenport [9] originally suggested an exponential expression as the root-coherence function of the along-wind turbulence, which is given by(21)Cohjkfn=exp−2fnCzrzU¯zj+U¯zk,where fn is the nth frequency, rz is the vertical separation between points Pj0,0,zj and Pk0,0,zk, and Cz is the vertical decay constant that controls the vertical correlation between turbulence at the two points. Some building codes propose a vertical decay constant of 8; whereas Davenport [9], Strømmen [10], Dyrbye and Hansen [11], and Solari [12] propose a vertical decay constant equal to 7, 9, 10, and 11.5, respectively.As mentioned earlier, (21) has the advantage of simplicity, but incorporates two inconsistencies [11]: (1) the function is positive for any separation, which is in conflict with the definition of the longitudinal turbulence component with a zero mean; and (2) the function approaches unity for small frequencies, which is not true for separations of the same order of magnitude or even larger than the average size of gusts. To solve the two inconsistencies mentioned above, Krenk [13] suggested a modified exponential expression as the root-coherence function, which is given by(22)Cohjkfn=1−f∗CzrzU¯zj+U¯zkexp−2f∗CzrzU¯zj+U¯zk,where(23)f∗=fn2+U¯zj+U¯zk2πLuzj+Luzk2,and Luzj is the integral length scale at height zj, which is defined in Appendix A.On the other hand, the IEC root-coherence function [20] is usually used for offshore wind turbines, and is given by(24)Cohjkfn=exp−122fnrzU¯zj+U¯zk2+0.12rzLc2,where Lc is a coherence scale parameter approximately equal to 340.2 m.Forzo=0.01m, U¯zr=32m/s, and U¯zr=55m/s, Figure 2 shows a comparison among the three different models for the root-coherence function at three different points P10,0,10m, P20,0,30m, and P30,0,100m; where different vertical decay constants ranging from 5 to 11.5 were used in the Davenport root-coherence function (21), whereas a vertical decay constant of 5 was used in the Krenk root-coherence function (22).Figure 2
Comparison of root-coherence functions for along-wind loads.
(a)(b)(c)(d)tUsing full-scale measurements with vertical separations of the order of 10–20 m, Hansen and Krenk [14] determined that a vertical decay constant of 5 in the modified exponential format proposed by Krenk [13] corresponds to a vertical decay constant of 7.5 in the exponential format proposed by Davenport [9], i.e., close to the vertical decay constant of 8 used by some building codes. For separations greater than 20 m, Figure 2 also shows the same pattern, whereas the IEC model [20] corresponds to a vertical decay constant approximately equal to 11.5 in (21). Figure 2 shows that different values of U¯zr do not significantly modify the vertical correlation discussed above. Therefore, it is highly recommended to correlate along-wind loads by using Cz=5 in (22).


### 3.2. Across-Wind Loads

According to Liang et al. [1], the root-coherence function for across-wind loads on rectangular tall buildings within the spectral peaks can be approximated by the following empirical equation:(25)Cohjkfs=exp−rzBΔ2,where fs is the vortex shedding frequency, rz is the vertical separation between points Pj0,0,zj and Pk0,0,zk, B is the width of the windward side, and Δ is a parameter related to the side ratio of the building: for 1/4≤D/B≤1 and Δ=5.56, whereas for 2≤D/B≤4 and Δ=7.7. However, substitution of (25) into (16) leads to a negative definite Hermitian coherence matrix that cannot be decomposed through the Cholesky factorization.As Tsukagoshi et al. [15] suggested, the root-coherence function proposed by Davenport [9] could be used to correlate across-wind loads. However, they did not justify the value of the vertical decay constant used in the simulation process. Accordingly, a comparison of the vertical coherence function between (21) and (25) within the spectral peaks could solve the problem. Unlike Equation (30), Equation (26) depends on the frequency content of the spectra; however, the values of the vertical decay constants that fit (25) are unknown.Assuming thatzo=0.01m, U¯zr=32m/s, H=200m, and B=30m, Figure 3 compares (25) with the Pearson correlation coefficients pjk, for different values of the vertical decay constant, which were computed once the across-wind loads were simulated by using (19) and (21).Figure 3
Vertical correlation for across-wind forces: B/H = 0.15.
(a)(b)The lack of aB/H=0.15 single value of Cz that perfectly matches (25) is shown in Figure 3, perhaps because both are processes of different nature. For this reason, a proposal of vertical decay constants not only for different values of D/B but also for different values of H seems to be a better option.(25) is valid within spectral peaks. Therefore, the average vortex shedding frequency between points Pj0,0,zj and Pk0,0,zk is given by(26)fs,jk=StU¯zj+U¯zk2B,where St is the Strouhal number. If (26) is substituted into (21), what follows is that the root-coherence function at the vortex shedding frequency is given by(27)Cohjkfs=exp−StCzrzB.The values ofCz in (27) that fit (25) were determined by assuming 6 different values of D/B ratio and 6 different values of B, that is, a total of 36 comparative graphs were generated for across-wind loads, 4 of which are shown in Figures 4 and 5. Based on the 36 comparative graphs, the values of Cz shown in Figure 6 are proposed to simulate across-wind loads on rectangular tall buildings by using (21) in the SR method based on the Cholesky decomposition of lagged coherency matrix.Figure 4
Vertical correlation at vortex shedding frequency for across-wind forces: D/B = 1.
(a)(b)Figure 5
Vertical correlation at vortex shedding frequency for across-wind forces: D/B = 2.
(a)(b)Figure 6
Proposed vertical decay constants for across-wind loads on rectangular tall buildings.
(a)(b)


### 3.3. Wind-Induced Torque

According to Liang et al. [2], the root-coherence function for wind-induced torque on rectangular tall buildings within the spectral peaks can be approximated by the following empirical equation:(28)Cohjkfs=cosΔ1rzBexp−rzBΔ22,where fs is the vortex shedding frequency, rz is the vertical separation between points Pj0,0,zj and Pk0,0,zk, B is the width of the windward side, whereas Δ1 and Δ2 are the parameters related to the side ratio of the building (see Table 2). However, substitution of (28) into (16) leads to a negative definite Hermitian coherence matrix that cannot be decomposed through the Cholesky factorization.Table 2
Suggested values of the parametersΔ1 and Δ2 [2].
D/B1/41/31/21.02–4Δ10000.55Δ21.672.55.05.567.7Assuming thatzo=0.01m, U¯zr=32m/s, H=200m, and B=30m, Figure 7 compares (28) with the Pearson correlation coefficients pjk, for different values of the vertical decay constant, which were computed once the wind-induced torque was simulated by using (19) and (21).Figure 7
Vertical correlation for wind-induced torque: B/H = 0.15.
(a)(b)B/H=0.15Similarly to across-wind loads, the root-coherence function at the vortex shedding frequency for wind-induced torque is given by (27). The values of Cz in (27) that fit (28) were determined by assuming 7 different values of D/B ratio and 6 different values of B, that is, a total of 42 comparative graphs were generated for wind-induced torque, 10 of which are shown in Figure 891011 to 12. For 2≤D/B≤4, it is assumed that Δ1=0, in this way, (27) can be fitted into (28) because the cosine term is removed. Based on the 42 comparative graphs, the values of Cz shown in Figure 13, 14 are proposed to simulate the wind-induced torque on rectangular tall buildings by using (21) in the SR method based on the Cholesky decomposition of lagged coherency matrix.Figure 8
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1/4.
(a)(b)Figure 9
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1/3.
(a)(b)Figure 10
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1/2.
(a)(b)Figure 11
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1.
(a)(b)Figure 12
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 2.
(a)(b)Figure 13
Proposed vertical decay constants for wind-induced torque on rectangular tall buildings: D/B = 1/4 and D/B = 1/3.
(a)(b)Figure 14
Proposed vertical decay constants for wind-induced torque on rectangular tall buildings2≦D/B≦4.
(a)(b)Based on Figure15, Table 3 shows the values of xn,yn, βn, and rn for the nth shear frame. On the other hand, Table 4 shows the first period of vibration of the buildings in each direction.Figure 15
Lateral resisting system of theN-story shear building: 2D plan view.Table 3
Cartesian coordinates, angles, and projected distances of all shear frames.
n12345678Frame1x2x3x4x1y2y3y4yx (m)0000−15−5515y (m)−15−55150000β (rad)0000π/2π/2π/2π/2r (m)155−5−15−15-5515Table 4
First period [s] of theN-story shear building in each direction: D/B = 1.
HBDNTranslational inx-directionTranslational iny-directionTorsional2151.311.311.014302.612.612.026453.923.923.038605.225.224.04It was assumed that the building is located in the city of Cancun, México, whereρa=1.2161kg/m3,λ=21.10°, and zo=0.01m. According to the Mexican wind load code [45], U¯zr=32m/s for a 10-year return period, which is related to the serviceability limit state of the building. For the buffeting response, the aerodynamic admittance function proposed by Castro et al. [16] was used to attenuate the power spectra of the along-wind loads (see Appendix A). The aerodynamic coefficients and vertical decay constants are shown in Table 5, whereas the Strouhal numbers are St=0.084 and St=0.062 for across-wind loads and wind-induced torque, respectively.Table 5
Aerodynamics coefficients and vertical decay constants for the wind loads:D/B = 1.
HBDH (m)Aerodynamic coefficientsCzCDCLCMKrenk´s model for along-wind loadsDavenport´s model for across-wind loadsDavenport´s model for wind-induced torque2601.180.4040.07750.700.9441201.300.4040.07751.401.8061801.400.4040.07752.102.7082401.480.4040.07752.803.60For H/BD=8, Figures 16, 17, 18, 19, and 20 show a well-fitting simulation for the standard deviations, PSD functions, Pearson correlation coefficients, and wind-induced loads, among other parameters.Figure 16
Longitudinal turbulence parameters along the height H/sqrt(BD) = 8.
(a)(b)(c)


## 3.1. Along-Wind Loads

On a purely empirical basis, Davenport [9] originally suggested an exponential expression as the root-coherence function of the along-wind turbulence, which is given by(21)Cohjkfn=exp−2fnCzrzU¯zj+U¯zk,where fn is the nth frequency, rz is the vertical separation between points Pj0,0,zj and Pk0,0,zk, and Cz is the vertical decay constant that controls the vertical correlation between turbulence at the two points. Some building codes propose a vertical decay constant of 8; whereas Davenport [9], Strømmen [10], Dyrbye and Hansen [11], and Solari [12] propose a vertical decay constant equal to 7, 9, 10, and 11.5, respectively.As mentioned earlier, (21) has the advantage of simplicity, but incorporates two inconsistencies [11]: (1) the function is positive for any separation, which is in conflict with the definition of the longitudinal turbulence component with a zero mean; and (2) the function approaches unity for small frequencies, which is not true for separations of the same order of magnitude or even larger than the average size of gusts. To solve the two inconsistencies mentioned above, Krenk [13] suggested a modified exponential expression as the root-coherence function, which is given by(22)Cohjkfn=1−f∗CzrzU¯zj+U¯zkexp−2f∗CzrzU¯zj+U¯zk,where(23)f∗=fn2+U¯zj+U¯zk2πLuzj+Luzk2,and Luzj is the integral length scale at height zj, which is defined in Appendix A.On the other hand, the IEC root-coherence function [20] is usually used for offshore wind turbines, and is given by(24)Cohjkfn=exp−122fnrzU¯zj+U¯zk2+0.12rzLc2,where Lc is a coherence scale parameter approximately equal to 340.2 m.Forzo=0.01m, U¯zr=32m/s, and U¯zr=55m/s, Figure 2 shows a comparison among the three different models for the root-coherence function at three different points P10,0,10m, P20,0,30m, and P30,0,100m; where different vertical decay constants ranging from 5 to 11.5 were used in the Davenport root-coherence function (21), whereas a vertical decay constant of 5 was used in the Krenk root-coherence function (22).Figure 2
Comparison of root-coherence functions for along-wind loads.
(a)(b)(c)(d)tUsing full-scale measurements with vertical separations of the order of 10–20 m, Hansen and Krenk [14] determined that a vertical decay constant of 5 in the modified exponential format proposed by Krenk [13] corresponds to a vertical decay constant of 7.5 in the exponential format proposed by Davenport [9], i.e., close to the vertical decay constant of 8 used by some building codes. For separations greater than 20 m, Figure 2 also shows the same pattern, whereas the IEC model [20] corresponds to a vertical decay constant approximately equal to 11.5 in (21). Figure 2 shows that different values of U¯zr do not significantly modify the vertical correlation discussed above. Therefore, it is highly recommended to correlate along-wind loads by using Cz=5 in (22).


## 3.2. Across-Wind Loads

According to Liang et al. [1], the root-coherence function for across-wind loads on rectangular tall buildings within the spectral peaks can be approximated by the following empirical equation:(25)Cohjkfs=exp−rzBΔ2,where fs is the vortex shedding frequency, rz is the vertical separation between points Pj0,0,zj and Pk0,0,zk, B is the width of the windward side, and Δ is a parameter related to the side ratio of the building: for 1/4≤D/B≤1 and Δ=5.56, whereas for 2≤D/B≤4 and Δ=7.7. However, substitution of (25) into (16) leads to a negative definite Hermitian coherence matrix that cannot be decomposed through the Cholesky factorization.As Tsukagoshi et al. [15] suggested, the root-coherence function proposed by Davenport [9] could be used to correlate across-wind loads. However, they did not justify the value of the vertical decay constant used in the simulation process. Accordingly, a comparison of the vertical coherence function between (21) and (25) within the spectral peaks could solve the problem. Unlike Equation (30), Equation (26) depends on the frequency content of the spectra; however, the values of the vertical decay constants that fit (25) are unknown.Assuming thatzo=0.01m, U¯zr=32m/s, H=200m, and B=30m, Figure 3 compares (25) with the Pearson correlation coefficients pjk, for different values of the vertical decay constant, which were computed once the across-wind loads were simulated by using (19) and (21).Figure 3
Vertical correlation for across-wind forces: B/H = 0.15.
(a)(b)The lack of aB/H=0.15 single value of Cz that perfectly matches (25) is shown in Figure 3, perhaps because both are processes of different nature. For this reason, a proposal of vertical decay constants not only for different values of D/B but also for different values of H seems to be a better option.(25) is valid within spectral peaks. Therefore, the average vortex shedding frequency between points Pj0,0,zj and Pk0,0,zk is given by(26)fs,jk=StU¯zj+U¯zk2B,where St is the Strouhal number. If (26) is substituted into (21), what follows is that the root-coherence function at the vortex shedding frequency is given by(27)Cohjkfs=exp−StCzrzB.The values ofCz in (27) that fit (25) were determined by assuming 6 different values of D/B ratio and 6 different values of B, that is, a total of 36 comparative graphs were generated for across-wind loads, 4 of which are shown in Figures 4 and 5. Based on the 36 comparative graphs, the values of Cz shown in Figure 6 are proposed to simulate across-wind loads on rectangular tall buildings by using (21) in the SR method based on the Cholesky decomposition of lagged coherency matrix.Figure 4
Vertical correlation at vortex shedding frequency for across-wind forces: D/B = 1.
(a)(b)Figure 5
Vertical correlation at vortex shedding frequency for across-wind forces: D/B = 2.
(a)(b)Figure 6
Proposed vertical decay constants for across-wind loads on rectangular tall buildings.
(a)(b)


## 3.3. Wind-Induced Torque

According to Liang et al. [2], the root-coherence function for wind-induced torque on rectangular tall buildings within the spectral peaks can be approximated by the following empirical equation:(28)Cohjkfs=cosΔ1rzBexp−rzBΔ22,where fs is the vortex shedding frequency, rz is the vertical separation between points Pj0,0,zj and Pk0,0,zk, B is the width of the windward side, whereas Δ1 and Δ2 are the parameters related to the side ratio of the building (see Table 2). However, substitution of (28) into (16) leads to a negative definite Hermitian coherence matrix that cannot be decomposed through the Cholesky factorization.Table 2
Suggested values of the parametersΔ1 and Δ2 [2].
D/B1/41/31/21.02–4Δ10000.55Δ21.672.55.05.567.7Assuming thatzo=0.01m, U¯zr=32m/s, H=200m, and B=30m, Figure 7 compares (28) with the Pearson correlation coefficients pjk, for different values of the vertical decay constant, which were computed once the wind-induced torque was simulated by using (19) and (21).Figure 7
Vertical correlation for wind-induced torque: B/H = 0.15.
(a)(b)B/H=0.15Similarly to across-wind loads, the root-coherence function at the vortex shedding frequency for wind-induced torque is given by (27). The values of Cz in (27) that fit (28) were determined by assuming 7 different values of D/B ratio and 6 different values of B, that is, a total of 42 comparative graphs were generated for wind-induced torque, 10 of which are shown in Figure 891011 to 12. For 2≤D/B≤4, it is assumed that Δ1=0, in this way, (27) can be fitted into (28) because the cosine term is removed. Based on the 42 comparative graphs, the values of Cz shown in Figure 13, 14 are proposed to simulate the wind-induced torque on rectangular tall buildings by using (21) in the SR method based on the Cholesky decomposition of lagged coherency matrix.Figure 8
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1/4.
(a)(b)Figure 9
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1/3.
(a)(b)Figure 10
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1/2.
(a)(b)Figure 11
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 1.
(a)(b)Figure 12
Vertical correlation at vortex shedding frequency for wind-induced torque: D/B = 2.
(a)(b)Figure 13
Proposed vertical decay constants for wind-induced torque on rectangular tall buildings: D/B = 1/4 and D/B = 1/3.
(a)(b)Figure 14
Proposed vertical decay constants for wind-induced torque on rectangular tall buildings2≦D/B≦4.
(a)(b)Based on Figure15, Table 3 shows the values of xn,yn, βn, and rn for the nth shear frame. On the other hand, Table 4 shows the first period of vibration of the buildings in each direction.Figure 15
Lateral resisting system of theN-story shear building: 2D plan view.Table 3
Cartesian coordinates, angles, and projected distances of all shear frames.
n12345678Frame1x2x3x4x1y2y3y4yx (m)0000−15−5515y (m)−15−55150000β (rad)0000π/2π/2π/2π/2r (m)155−5−15−15-5515Table 4
First period [s] of theN-story shear building in each direction: D/B = 1.
HBDNTranslational inx-directionTranslational iny-directionTorsional2151.311.311.014302.612.612.026453.923.923.038605.225.224.04It was assumed that the building is located in the city of Cancun, México, whereρa=1.2161kg/m3,λ=21.10°, and zo=0.01m. According to the Mexican wind load code [45], U¯zr=32m/s for a 10-year return period, which is related to the serviceability limit state of the building. For the buffeting response, the aerodynamic admittance function proposed by Castro et al. [16] was used to attenuate the power spectra of the along-wind loads (see Appendix A). The aerodynamic coefficients and vertical decay constants are shown in Table 5, whereas the Strouhal numbers are St=0.084 and St=0.062 for across-wind loads and wind-induced torque, respectively.Table 5
Aerodynamics coefficients and vertical decay constants for the wind loads:D/B = 1.
HBDH (m)Aerodynamic coefficientsCzCDCLCMKrenk´s model for along-wind loadsDavenport´s model for across-wind loadsDavenport´s model for wind-induced torque2601.180.4040.07750.700.9441201.300.4040.07751.401.8061801.400.4040.07752.102.7082401.480.4040.07752.803.60For H/BD=8, Figures 16, 17, 18, 19, and 20 show a well-fitting simulation for the standard deviations, PSD functions, Pearson correlation coefficients, and wind-induced loads, among other parameters.Figure 16
Longitudinal turbulence parameters along the height H/sqrt(BD) = 8.
(a)(b)(c)


## 4. Wind-Induced Vibrations in Time Domain

Let us consider1/2≤D/B≤4 a three-dimensional N-story building, where the diaphragm is assumed to be rigid and therefore there will be three degrees of freedom per floor. Assuming a Wilson-Penzien damping [40], the equations of motion can be grouped in the following matrix system [41]:(29)MxMyJu¨xu¨yθ¨z+CxxCxyCxθCyxCyyCyθCθxCθyCθθu˙xu˙yθ˙z+KxxKxyKxθKyxKyyKyθKθxKθyKθθuxuyθz=Fxz,tFyz,tMzz,t,where(30)MxMym1⋱mN,JJ1⋱JN,CxxCxyCxθCyxCyyCyθCθxCθyCθθ∑i=13N2ξiwiMi∗mg,i,Mi∗=MxMyJ∅x,i∅y,i∅θ,i∅x,i∅y,i∅θ,iMxMyJ,mg,i∅x,i∅y,i∅θMxMyJ∅x,i∅y,i∅θ,i,Kxx=∑n=1MmKD,nCos2βn,Kyy=∑n=1NmKD,nSin2βn,Kxy=Kθx=∑n=1NmKD,nCosβnSinβn,Kxy=Kyx=∑n=1NmKD,nrnCosβn,Kyθ=Kθy∑n=1NmKD,nrnSinβn,Kθθ∑n=1NmKD,nrn2,rn=xnsinβn−yncosβn,Nm is the total number of frames of the building; n=1,2,3,…,Nm; j=1,2,3,…,N; mj is the lumped mass at the jth story of the building; Jj is the mass moment of inertia of the jth floor about the vertical axis passing through the center of mass (C.M.); ξi is the ith modal damping ratio; wi is the angular frequency of the ith mode of vibration; mg,i is the generalized mass corresponding to the ith mode of vibration; ϕx,iis the mode shape vector corresponding to the ith translational mode in x-direction; ϕy,i is the mode shape vector corresponding to the ith translational mode in y-direction; ϕθ,i is the mode shape vector corresponding to the ith torsional mode; KD,n is the condensed stiffness matrix of the nth frame corresponding to the lateral degrees of freedom; rn is the projected distance from the nth frame to the center of the mass of the rigid diaphragm; xn,yn are the Cartesian coordinates of the center of the mass of the nth frame, assuming that the origin of the Cartesian coordinate system is the center of the mass of the rigid diaphragm; βn is the angle between the nth frame and the x-axis; ux is a vector containing the lateral displacements of the rigid floor diaphragms in the x-direction; uy is a vector containing the lateral displacements of the rigid floor diaphragms in the y-direction; θz is a vector containing the angular displacements of the rigid floor diaphragms about the z-axis; Fxz,t is a vector containing the wind-induced forces in the x-direction; Fyz,t is a vector containing the wind-induced forces in the y-direction; and Mzz,t is a vector containing the wind-induced torque about the z-axis. The matrix equation (34) can be solved using a numerical integration method such as the state space method [42].For standard structures, the across-wind and torsional responses are generally less significant than the along-wind response. However, for structures with significant height, the across-wind and torsional responses can become more significant [43]. According to the National Research Council of Italy [43], the effects of across-wind and torsional actions should be assessed if H/BD≥3. In this paper, the analyses were performed for a square plan building (D/B=1) and 4 different values of H/BD. Thereby, the effects of vertical correlation on along-wind, across-wind, and torsional vibrations can be assessed for different aspect ratios.The mass per unit length of the building is equal to180000kg/m, which means that mkk=1,2,3,…,N−1, mN, Jkk=1,2,3,…,N−1, and JN are 720000kg, 360000kg, 1.08×108kgm2, and 5.4×107kgm2, respectively. Based on Figure 15, the lateral resisting system in each translational direction consists of 4 uniform shear frames, that is, Nm=8. In this idealization, the beams and floor systems are rigid (infinitely stiff) in flexure, and several factors are neglected: the axial deformation of the beams and columns, and the effect of axial force on the stiffness of the columns. Assuming a story height equal to 4 m, the lateral stiffness of each column is 9.5125×107N/m; therefore, the story stiffness for each translational direction is equal to 1.522×109N/m. In wind engineering, the total damping ratio is composed of the structural part and the aerodynamic part, where the aerodynamic damping is caused by the interaction between the motion of a structure and motion of the air around it. In this numerical example, it was assumed that ξii=1,2,3,…,3N is equal to 0.01 [44].Figure 17
Standard deviations along the height H/sqrt(BD) = 8.
(a)(b)(c)Figure 18
Power-spectral density functions at the rooftop H/sqrt(BD) = 8.
(a)(b)(c)Figure 19
Vertical coherence of wind-induced forces H/sqrt(BD) = 8.
(a)(b)(c)Figure 20
Simulation of wind-induced loads in time domain H/sqrt(BD) = 8.
(a)(b)(c)Figures21, 22, and 23 compare the co H/BD=8 related H/BD=8 rooftop H/BD=8 response in time H/BD=8 domain withH/BD=8 the uncorrelated rooftop response, that is, if the vertical coherence function is neglected by reducing (19) to gjt=∑n=1m2ΔfSgzj,fncos2πfnt+ϕn. Accordingly, Table 6 shows the root-mean-square (RMS) and peak values of the rooftop dynamic response for both uncorrelated and correlated wind loads.Figure 21
Rooftop displacements of the shear buildings in time domain. (a) H/√BD=2, (b) H/√ BD=4. (c) H/√BD=6. (d) H/√BD=8.
(a)(b)(c)(d)Figure 22
Rooftop velocities of the shear buildings in time domain. (a) H/√BD=2, (b) H/√ BD=4. (c) H/√BD=6. (d) H/√BD=8.
(a)(b)(c)(d)Figure 23
Rooftop accelerations of the shear buildings in time domain. (a) H/√BD=2, (b) H/√ BD=4. (c) H/√BD=6. (d) H/√BD=8.
(a)(b)(c)(d)Table 6
Effect of vertical correlation on the rooftop dynamic response of the shear buildings: time domain.
Uncorrelated/CorrelatedDirectionH/BD2468Peak displacement (%)Along-wind (y-direction)−11.94+15.33−1.69+10.97Across-wind (x-direction)−67.79−75.28−81.23−77.75Torsion aboutz-axis−67.16−71.79−73.44−77.42RMS acceleration (%)Along-wind (y-direction)+109.33+67.26+85.42+61.77Across-wind (x-direction)+15.57−0.055−33.18−29.41Torsion aboutz-axis+242.67+122.92+71.64+122.09Peak acceleration (%)Along-wind (y-direction)+25.47+42.40+26.70+34.94Across-wind (x-direction)+51.48+14.58−25.08−36.45Torsion aboutz-axis+236.27+96.27+97.47+116.04Peak displacements are frequently used to assess wind-induced damage, whereas peak and root-mean-square (RMS) accelerations are commonly used for human comfort assessment to wind-induced motion [46]. Therefore, Table 6 shows the effects of the vertical correlation on the wind-induced responses for different aspect ratios. Generally speaking, Table 6 shows that the along-wind peak displacements are slightly modified when vertical correlation is neglected, that is, in the range of 2 to 15%; on the contrary, the peak displacements of both across-wind and torsional directions decrease 67–81% for all aspect ratios. When the vertical is neglected, RMS values of along-wind accelerations increase 62–109%, whereas peak values of along-wind accelerations increase 25–42% for all aspect ratios. When the vertical correlation is neglected, both RMS and peak values of across-wind accelerations increase 15–51% for aspect ratios in the range of 2 to 4, whereas they decrease 25–36% for aspect ratios in the range of 6 to 8. On the other hand, RMS torsional accelerations increase 72–243% when the vertical correlation is neglected, whereas peak torsional accelerations increase 97–236% for all aspect ratios.


## 5. Wind-Induced Vibrations in Frequency Domain

According to Newland [47], the frequency response function matrix between the generalized structural displacements and the external forces of a multi degree of freedom (MDOF) system can be obtained using the following equation:(31)Bw=K−w2M+iwC−1,where K,M, and C are the stiffness, mass, and damping matrices, respectively; w is the angular frequency and i is the imaginary unit. Then, the displacement and acceleration power-spectral density (PSD) matrices are defined, respectively, as(32)Suuw=Bw∗SggwBw,Su¨u¨w=w4Suuw,where the ∗ superscript denotes complex matrix conjugation and Sggw is the cross-spectral density matrix of the wind forces, which is given in (11). Next, the variance of the response is obtained by integrating the response auto-spectra populating the main diagonal elements of the response PSD matrix on the frequency axis up to a maximum (cut-off) frequency wm above which the energy of the underlying processes is negligible. In this way, the RMS displacement and RMS acceleration of the jth floor are obtained, respectively, as(33)σuj=∫0wmSujujwdw,σu¨j=∫0wmSu¨ju¨jwdw,where Sujujw and Su¨ju¨jw are the jth elements of the main diagonal of Suuw and Su¨u¨w matrices, respectively. For both across-wind and torsional-wind loads, the peak displacement and peak acceleration of the jth floor can be estimated, respectively, by the expressions(34)peakuj=kpσuj,peaku¨j=kpσu¨j,where kp is the peak factor estimated by the widely used empirical formula [48] as follows:(35)kp≈2lnTs/T1∗+0.5772lnTs/T1∗,where T1∗ is the fundamental period of vibration of the structure in the direction of analysis and Ts is the time duration of exposure to the wind action, which is usually equal to 600 s.As already mentioned above, both across-wind and torsional-wind loads cannot be simulated in time domain by using the empirical root-coherence functions proposed by Liang et al. [1, 2] because these functions lead to a negative definite Hermitian coherence matrix that cannot be decomposed through the Cholesky factorization. However, the empirical root-coherence functions proposed by Liang et al. [1, 2] can be effectively used to compute the elastic wind-induced vibrations in the frequency domain by using Equations (50) and (34). In this way, the vertical decay constants proposed in this paper (see Figures 6, 13, and 14) can be validated by comparing the correlated time domain response with the correlated frequency domain response resulting from using the empirical coherence functions proposed by Liang et al [1, 2]. Accordingly, Table 7 shows a close approximation between the time domain and frequency domain responses.Table 7
Effect of the empirical coherence function on elastic dynamic response of the shear buildings.
HBDRooftop responseAcross-wind response [m, m/s2]Wind-induced torsional response [rad, rad/s2]Time domain (davenport)Frequency domain (liang)Time domain (davenport)Frequency domain (liang)2RMS displacement0.00600.00620.000170.00017Peak displacement0.02080.02280.000670.00062RMS acceleration0.12780.12410.00750.0067Peak acceleration0.41760.45460.02840.02154RMS displacement0.03180.03790.00130.0016Peak displacement0.09870.13150.00390.0057RMS acceleration0.18130.20200.01440.0154Peak acceleration0.58030.70140.05630.05446RMS displacement0.10450.10990.00400.0050Peak displacement0.33040.36870.01280.0170RMS acceleration0.26430.26520.02010.0215Peak acceleration0.86160.88960.07110.07378RMS displacement0.21580.26960.00380.0047Peak displacement0.74650.88090.01240.0223RMS acceleration0.29850.36820.01630.0171Peak acceleration1.13131.20310.05860.0570Generally speaking, Table7 shows that RMS elastic response decreases in the range of 0.3 to 20% when the time domain analysis is used in conjunction with the proposed vertical decay constants (see Figures 6, 13, and 14). On the other hand, the peak elastic response decreases in the range of 3 to 44% because the peak factor shown in (35) is just an approximation based on along-wind response; therefore, the RMS elastic response is the recommended one to be taken as a validation parameter for the vertical decay constants proposed in this paper. In this way, a time domain analysis has a wider field of application because it is not only useful for nonlinearity behavior, nonclassically damping, and detailed structural models in commercial software but also to compute precisely the peak elastic response compared to an analysis in the frequency domain where the peak factor may not be suitable for across-wind and torsional-wind responses.


## 6. Nonlinear Time History Analysis

As already mentioned above, the nonlinear behavior of a structure hinders the use of a frequency domain approach. Therefore, a nonlinear time history analysis will show the advantages of the vertical decay constants proposed in this paper to simulate the wind-induced vibrations considering the correlation along the height of the building.In this section, the 60-story shear building (see Figure15) is subjected simultaneously to the along-wind and across-wind loads shown in Figure 20, which are correlated along the height of the structure. For the nonlinear behavior, it was assumed a bilinear isotropic hardening ASTM A992 steel model considering a Young´s modulus equal to 199 GPA; a yield stress equal to 344.7 MPA; and a postyield story stiffness equal to 304.4 MN/m, that is, 20% of elastic story stiffness. Assuming a story height equal to 4 m, the yield story force and yield story displacement are 10.5 MN and 6.9 mm, respectively.The nonlinear matrix system was solved in MATLAB by using the Newmarkβ-method [42], in this way, the nonlinear displacements and hysteretic loops of the 60th floor and 5th floor are shown in Figures 24 and 25, respectively. In addition, Figure 26 shows the energy balance in order to check that the total energy (kinetic + dissipated + elastic strain + hysteresis) is equal to the input energy.Figure 24
Nonlinear response in time domain of the 60th floor H/sqrt(BD) = 8.
(a)(b)(c)Figure 25
Nonlinear response in time domain of the 5th floor H/sqrt(BD) = 8.
(a)(b)(c)Figure 26
Energy balance for the nonlinear time history analysis H/sqrt(BD) = 8.
(a)(b)For the along-wind response, the structural material yields from 1st story to the 31st story, whereas it remains elastic from story the 32nd story to the rooftop. On H/BD=8 the other hand, H/BD=8 for the across-wind response, the structural material yields from 1st story to the 50th story, whereas it remains elastic from the 51st story to the rooftop. In fact, Figure 26 clearly shows that the inputH/BD=8 energy related to the across-wind loads is much greater than the input energy related to the along-wind loads. Furthermore, the hysteresis energy in Figure 26(a) remains approximately constant over time because the along-wind loads are strictly “positive” most of the time. Therefore, the hysteresis behavior usually remains in the loading branches (see Figure 25(b)). On the contrary, Figure 25(c) shows multiple hysteresis loops related to the across-wind vibrations, which means that both hysteresis energy and dissipated energy increase over time (see Figure 26(b)).For the along-wind response at the rooftop, Figure24(a) shows that the peak displacement is equal to 68.7 cm, which is 1.37 times greater than that computed by a linear time history analysis (see Table 6). On the contrary, Figure 24(a) shows that the across-wind peak displacement at the rooftop is equal to 51.8 cm, that is, it decreases 30.6% compared to that computed by a linear time history analysis (see Table 6).


## 7. Conclusions

The spectral representation (SR) method is frequently used to simulate along-wind loads on tall buildings. However, it can be used to simulate any mean-zero multivariate and multidimensional process such as across-wind loads and wind-induced torque. However, scientific literature reveals that across-wind and wind-induced torque have only been simulated in time domain without considering the vertical correlation between the signals, which is physically incorrect. Therefore, it has not been possible to simulate both linear and nonlinear response induced by vortex shedding (across and torsional directions). For the particular case of nonlinear response, it is very important to compute the damage induced by the wind in the 3D space.As the first goal of this research, a review of the parameters involved to simulate along-wind, across-wind, and torsional-wind loads on rectangular tall buildings was carried out to unify the simulation process in order to compute the 3D wind-induced vibrations. In this way, the most important empirical models for the parameters of wind simulations are fully discussed. Accordingly, the following conclusions were obtained:(a)
For along-wind loads, the power-spectral density function proposed in ESDU adequately represents the power of the turbulent component in all the frequency content. Furthermore, it is highly recommended to correlate along-wind loads by using a vertical decay constant equal to 5 in the empirical coherence function proposed by Krenk, which is not encumbered by the two inconsistencies related to the empirical function proposed by Davenport.(b)
For across-wind and torsional-wind loads, the power-spectral density functions proposed by Liang adequately represent the power of both types of wind loads on rectangular tall buildings with various side ratios. However, the empirical coherence functions proposed by Liang cannot be used in the SR method because they do not depend on the frequency content of the spectra and lead to a negative definite coherence matrix that cannot be factorized through the Cholesky decomposition.As the second goal of this research, a comparative analysis between the Liang coherence model and the Davenport coherence model was carried out to propose vertical decay constants for various side ratios at normal attack angle, which is the most important contribution of this paper because it solves the problem of vertical correlation in the simulation of across-wind and torsional-wind loads in time domain. Accordingly, the following conclusions were obtained:(c)
There is not a single value of the vertical decay constant in Davenport’s empirical model that perfectly matches Liang coherence function, particularly whenH decreases and B increases.(d)
The Davenport coherence function fits well the Liang coherence function asH increases and B decreases.As the third goal of this research, the effect of vertical correlation on along-wind, across-wind, and wind-induced torsional responses were assessed in the time domain for shear buildings with different aspect ratios by using a three‐dimensional model with two translational and one torsional degree of freedom for each floor. Furthermore, the proposed vertical decay constants were validated by comparing the elastic response between frequency domain and time domain approaches. Accordingly, the following conclusions were obtained:(e)
Generally speaking, the results show that the peak displacements are significantly underestimated for both across-wind and torsional directions when the vertical correlation is neglected. On the contrary, the along-wind peak displacements are slightly modified.(f)
When the vertical correlation is neglected, across-wind acceleration increases for low-rise buildings, whereas it decreases for high-rise buildings. On the other hand, torsional accelerations increase significantly when the vertical correlation is neglected.(g)
Generally speaking, the RMS elastic response decreases in the range of 0.3 to 20% when the time domain analysis is used in conjunction with the proposed vertical decay constants for both across-wind and torsional directions.(h)
For the elastic RMS response considering vertical correlation, the difference between time domain and frequency domain approaches was mostly negligible.(i)
The peak factor in the frequency domain may not be suitable for across-wind and torsional-wind elastic responses. Therefore, a time domain analysis has a wider field of application.As the final goal of this research, the advantages of time domain simulation were shown by performing a nonlinear time history analysis for a 60-story shear building considering a bilinear isotropic material hardening model in both translational directions. Accordingly, the following conclusions were obtained:(j)
For the along-wind direction, the structural material yields from 1st story to the 31st story, whereas it yields from 1st story to the 50th story for the across-wind direction.(k)
The along-wind peak displacement at rooftop increases 37.3% when a nonlinear time history analysis is performed. On the contrary, the across-wind peak displacement decreases 30.6% compared to that computed by a linear time history analysis.Based on contributions of this research, we believe that structural engineers will be able to compute accurately the 3D wind-induced response of rectangular tall buildings for specific problems where frequency domain analysis is not applicable (nonlinearity behavior, nonclassically damping, and detailed structural models in commercial software), particularly for across-wind and torsional responses. For future work, the 3D wind-induced vibration of high-rise buildings will be studied in time domain considering nonlinear behavior and passive energy dissipation devices.



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*Source: 7283610-2022-10-07.xml*