# Evaluation of Wind-Induced Response Bounds of High-Rise Buildings Based on a Nonrandom Interval Analysis Method

Authors: Xianglei Wei; An Xu; Ruohong Zhao
Journal: Shock and Vibration
 (2018)

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## Abstract

The traditional wind-induced response analysis of high-rise buildings conventionally considers the wind load as a stationary stochastic process. That is, for a certain wind direction angle, the reference wind speed (usually refers to the mean wind speed at the building height) is assumed to be a constant corresponding to a certain return period. Combined with the recorded data in wind tunnel test, the structural response can be computed using the random vibration theory. However, in the actual typhoon process, the average wind speed is usually time-variant. This paper combines the interval process model and the nonrandom vibration analysis method with the wind tunnel test and proposes a method for estimating the response boundary of the high-rise buildings under nonstationary wind loads. With the given upper and lower bounds of time-variant wind excitation, this method can provide an effective calculation tool for estimating wind-induced vibration bounds for high-rise buildings under nonstationary wind load. The Guangzhou East tower, which is 530 m high and the highest supertall building in Guangzhou, China, was taken as an example to show the effectiveness of the method. The obtained boundary response can help disaster prevention and control during the passage of typhoons.


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## Body

## 1. Introduction

The conventional methods of wind-induced vibration of tall buildings are based on the basic assumption that wind loads are stationary random processes [1–4]. That is, assuming the reference wind speed of a certain wind direction is a constant. Combined with wind tunnel test data, this assumption can be used to calculate structural response and provide the peak response for a structural design [5]. An actual typhoon has obvious nonstationary characteristics. The average wind speed at the reference height of a building is time-varying rather than constant. The conventional structural wind-induced response analysis framework cannot effectively predict structural response at each moment when a typhoon passes. In actual operation, this nonrandom vibration analysis method is used to predict typhoon wind speed from a small sample and obtain the upper and lower boundaries of the typhoon. Combined with high-frequency force balance (HFFB) wind tunnel test data, the boundary values of structural response at each time instant can be predicted, and the boundary response of a building can be estimated to guide disaster prevention and control.During a real typhoon, average wind speed at the reference height of a building is time-variant. Researchers have developed methods to estimate the response boundaries of a system subjected to nonstationary loads. Qiu and Elishakoff proposed a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters [6]. Chen et al. [7] proposed a method to solve the static displacement problem with interval parameter structure uncertainty. Qiu and Wang used parametric perturbation methods to evaluate the dynamic response of structures with uncertain but bounded parameters [8]. Moens and Vandepitte proposed a method of interval sensitivity evaluation in the framework of general numerical interval analysis [9]. Verhaeghe et al. [10] proposed the concept of interval domains, considering static analysis of a structure with interval uncertainty using the finite element method. With the development of interval convex model theory, Jiang et al. [11, 12] proposed a nonrandom vibration analysis method for nonprobabilistic convex models. It is only necessary to know the boundary excitation information at an arbitrary time point rather than the exact probability distribution. The solution system is a responsive interval under uncertain excitation. This method reduces the dependence on large sample sizes and considers the correlation between interval variables and provides a new nonlinear interval programming method, which can be used to deal with uncertain optimization problems when there is a dependency relationship among interval variables. Gao et al. [13] proposed a mixed probability and interval reliability analysis method to solve engineering problems. Considering that uncertain factors widely exist in practical engineering, Wang et al. [14–16] proposed a novel methodology of reliability-based multidisciplinary design optimization under hybrid interval and fuzzy uncertainties, and a dimension-wise method and its improvement for multidisciplinary interval uncertainty analysis, and the methods of hybrid time-variant reliability estimation for active control structures under aleatory and epistemic uncertainties. These methods have important value in solving the uncertainty problem in practical engineering.The HFFB technique is one of the most widely used techniques to evaluate effects of wind-induced vibration of tall buildings [17–22]. The nonrandom vibration theory of the interval process model is combined with HFFB wind tunnel tests to predict the boundary response, including displacement, velocity, and acceleration response, of high-rise buildings subjected to nonstationary wind loads. Through a case analysis of Guangzhou East Tower (GZET), the feasibility of the method is examined.


## 2. Methodology

### 2.1. Interval Process Model

For an uncertain processXt,t∈T, T is the parameter set of time instant t. For any time instant tii=1,2,...,n, if the possible values of Xti are within the interval XLt,XUt, the process Xt,t∈T is called an interval process model, denoted as XIt [23]. For the interval process model XIt, XUt, and XLt are the upper and lower bound functions, respectively. The middle point function of XIt and XCt, is defined as(1)XCt=XUt+XLt2.The half width functionXWt of the interval process model XIt is defined as(2)XWt=XUt−XLt2.The variation coefficient functionηXt of the interval process model XIt is defined as(3)ηXt=XWtXCt.For an interval processXIt, the self-correlation function between any two interval variables XIti and XItj at different times ti and tj is defined as rXXti,tj(4)rXXti,tj=1/m∑p=1mxip−XiCxjp−XjCWiXWjX,where WiX=1/m∑p=1mxip−XiC2 and WjX=1/m∑p=1mxjp−XjC2; xip and xjp are the values of the pth sample function at time ti and tj, respectively; and XiC and XjC are the values of the middle point function of XIt at time ti and tj, respectively.


### 2.2. Nonrandom Vibration Analysis Method

Based on the interval process model, Jiang et al. [23] proposed a nonrandom vibration analysis method. In this method, the time-varying uncertain excitation is treated as an interval process model, and the corresponding structural response is the interval form of the upper and lower boundaries. In practical applications, this nonrandom vibration analysis method, combined with HFFB technology, can estimate the structural response boundaries of tall buildings and give nonstationary reference wind speeds. This method requires only conventional wind tunnel tests which are carried out in a stationary wind field.


#### 2.2.1. Under Deterministic Excitation

As shown in Figure1, consider a simple single degree of freedom (SDOF) oscillator model, with mass m, spring stiffness k, damping coefficient c, and external force Ft. The motion equation can be given in the form as follows:(5)mz¨+cz˙+kz=Ft,or(6)z¨+2ζωnz˙+ωnz=Ftm,where z, z˙, and z¨ are the displacement, velocity, and acceleration response for the first mode in each direction of the system, respectively. The natural frequency ωn and damping ratio ζ are given by(7)ωn=km,cc=2mωn,ζ=ccc,where cc is the critical damping coefficient.Figure 1
SDOF spring-mass-damper vibration system.
(a)
(b)In this paper, the response of Equations (5) or (6) is solved using Duhamel’s integral. Herein, the piecewise linear interpolation function [24] is first used to approximately fit the sequence of Fti (denoted as Fi), where the rate of change of Ft within arbitrary period of time is supposed to be linear. In this way, Duhamel’s integral can be then applied to compute the response at time t=tj, j=1,2,3,…,n:(8)zj=ΔFjkΔtjΔtj−2ζωn+e−ζωnΔtj2ζωncosωdΔtj−ωd2−ζ2ωnωn2ωdsinωdΔtj+Fj−1k1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj+e−ζωnΔtjzj−1cosωdΔtj+z˙j−1+ζωnzj−1ωdsinωdΔtj,where ΔFj=Fj−Fj−1, Δtj=tj−tj−1, and ωd=ωn1−ζ2. The velocity z˙j at time t=tj can be calculated as follows:(9)z˙j=ΔFjkΔtj1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj+Fj−1ke−ζωnΔtjωn2ωdsinωdΔtj+e−ζωnΔtjz˙j−1cosωdΔtj−ζωnωdz˙j−1+ωnζzj−1sinωdΔtj.Equations (8) and (9) can be rewritten in the following form:(10)zj=AjΔFj+BjFj−1+Cjzj−1+Djz˙j−1,z˙j=A¯jΔFj+B¯jFj−1+C¯jzj−1+D¯jz˙j−1,where(11)Aj=1kΔtjΔtj−2ζωn+e−ζωnΔtj2ζωncosωdΔtj−ωd2−ζ2ωnωn2ωdsinωdΔtj,Bj=1k1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,Cj=e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,Dj=e−ζωnΔtj1ωdsinωdΔtj,A¯j=1kΔtj1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,B¯j=1ke−ζωnΔtjωn2ωdsinωdΔtj,C¯j=−e−ζωnΔtjωn2ωdsinωdΔtj,D¯j=e−ζωnΔtjcosωdΔtj−ζωnωdsinωdΔtj.Rewriting Equation (10) in matrix form gives(12)zjz˙j=AjBj−AjCjDjA¯jB¯j−A¯jC¯jD¯jFjFj−1zj−1z˙j−1.By using either Equations (10) or (12) repeatedly, the following recurrence sequences can be obtained:(13)FjFj−1zj−1z˙j−1=10000010000Aj−1Bj−1−Aj−1Cj−1Dj−10A¯j−1B¯j−1−A¯j−1C¯j−1D¯j−1FjFj−1Fj−2zj−2z˙j−2=Hj−1FjFj−1Fj−2zj−2z˙j−2,FjFj−1Fj−2zj−2z˙j−2=10000001000000100000Aj−2Bj−2−Aj−2Cj−2Dj−200A¯j−2B¯j−2−A¯j−2C¯j−2D¯j−2FjFj−1Fj−2Fj−3zj−3z˙j−3=Hj−2FjFj−1Fj−2Fj−3zj−3z˙j−3,FjFj−1⋮F1z1z˙1=10⋯000001⋯0000⋮⋮⋱00000001000000A1B1−A1B1D1000A¯1B¯1−A¯1B¯1D¯1FjFj−1⋮F1F0z0z˙0=H1FjFj−1⋮F1F0z0z˙0.Hence, there is:(14)FjFj−1zj−1z˙j−1=Hj−1,Hj−2,…,H1FjFj−1⋮F1F0z0z˙0.LetH∗=AjBj−AjCjDjA¯jB¯j−A¯jC¯jD¯j, H=H∗∏k=1j−1Hk, F=FjFj−1⋯F1F0Τ, and z0=z0z˙0Τ, then Equation (12) can be expressed as(15)zjz˙j=HFz0,where H is a 2×j+3 constant coefficient matrix, F is the vector of force sequence, and z0 is the initial condition.


#### 2.2.2. Under Uncertain Excitation

In nonrandom vibration analysis of the interval process model, the external excitation is treated as an interval process,FIt=FLt,FUt, and the autocorrelation function is rti,tj. FIt can be represented by an interval vector, FI=FjIFj−1I⋯F1IF0IT (Equation (16)). Therefore, Equation (15) can be rewritten by eliminating correlation between variables and transforming them into an independent interval processes F¯It:(16)zjz˙j=HFIz0.F
I
t is obtained through Cholesky decomposition, by eliminating the correlation between variables and transforming them into independent interval processes, F¯It, with a corresponding interval vector, F¯I. The following equation can be obtained [23]:(17)F¯I=L−1FI.The corresponding interval middle point vector isF¯C, for which the following equation can be obtained:(18)F¯C=L−1FC.The following equation can be obtained for the corresponding interval half width vectorF¯W:(19)F¯W=L−1FW,where L is a j+1×j+1 lower triangular matrix, L denotes the matrix whose entries are absolute values of corresponding ones in the matrix L.Through this treatment, Equation (16) can be rewritten as [23]:(20)zjz˙j=HLF¯Iz0=HFHZLF¯Iz0=HFLF¯I+HZz0,where HF and HZ are block matrices of H, with dimensions 2×j+1 and 2×2, respectively.DenoteδF¯W as(21)δF¯W=δjIF¯jW⋯δ1IF¯1Wδ0IF¯0WT,δiI=−1,1,i=0,1,…,j.LetF¯I=F¯C−F¯W,FC+F¯W. According to Equations (18)–(20), F¯I can be rewritten as(22)F¯I=F¯C+δF¯W.Substituting Equations (22), (18), and (19) in Equation (20) yields(23)zjz˙j=HFLF¯I+HZz0=HFLF¯C+HFLδF¯W+HZz0∈HFLF¯C+HZz0−HFLF¯W,HFLF¯C+HZz0+HFLF¯W=HFLL−1FC+HZz0−HFLL−1FW,HFLL−1FC+HZz0−HFLL−1FW=HFFC+HZz0−HFLL−1FW,HFFC+HZz0−HFLL−1FW.This implies that the upper and lower bounds of the displacement response at timet=tj are(24a)zjU=HF1FC+HZ1z0+HF1LL−1FW,(24b)zjL=HF1FC+HZ1z0−HF1LL−1FW.Similarly, the upper and lower bounds of velocity response at timet=tj can be obtained as(25a)z˙jU=HF2FC+HZ2z0+HF2LL−1FW,(25b)z˙jL=HF2FC+HZ2z0−HF2LL−1FW.Equation (5) can be rewritten as(26)z¨j=−cmz˙j−kmzj+Ftm.Also(27)z˙j=z˙j.Rewrite Equations (26) and (27) in matrix form:(28a)z˙jz¨j=01−km−cmzjz˙j+01mFt.Fromzj∈zjL,zjU=zjI,z˙j∈z˙jL,z˙jU=z˙jI,z¨j∈z¨jL,z¨jU=z¨jI,Ft∈FLt,FUt=FIt, where zjI, z˙jI and z¨jI are the interval displacement response, interval velocity response, and interval acceleration response, respectively.Equation (28a) can be rewritten as(28b)z˙jIz¨jI=01−km−cmzjIz˙jI+01mFIt.Equation (26) indicates that the upper and lower bounds of acceleration response can be obtained as(29a)z¨jL=−cmz˙jU−kmzjU+FLtm,(29b)z¨jU=cmz˙jL−kmzjL+FUtm,where FUt is the upper bound of excitation force and FLt is the lower bound of excitation force.


## 2.1. Interval Process Model

For an uncertain processXt,t∈T, T is the parameter set of time instant t. For any time instant tii=1,2,...,n, if the possible values of Xti are within the interval XLt,XUt, the process Xt,t∈T is called an interval process model, denoted as XIt [23]. For the interval process model XIt, XUt, and XLt are the upper and lower bound functions, respectively. The middle point function of XIt and XCt, is defined as(1)XCt=XUt+XLt2.The half width functionXWt of the interval process model XIt is defined as(2)XWt=XUt−XLt2.The variation coefficient functionηXt of the interval process model XIt is defined as(3)ηXt=XWtXCt.For an interval processXIt, the self-correlation function between any two interval variables XIti and XItj at different times ti and tj is defined as rXXti,tj(4)rXXti,tj=1/m∑p=1mxip−XiCxjp−XjCWiXWjX,where WiX=1/m∑p=1mxip−XiC2 and WjX=1/m∑p=1mxjp−XjC2; xip and xjp are the values of the pth sample function at time ti and tj, respectively; and XiC and XjC are the values of the middle point function of XIt at time ti and tj, respectively.


## 2.2. Nonrandom Vibration Analysis Method

Based on the interval process model, Jiang et al. [23] proposed a nonrandom vibration analysis method. In this method, the time-varying uncertain excitation is treated as an interval process model, and the corresponding structural response is the interval form of the upper and lower boundaries. In practical applications, this nonrandom vibration analysis method, combined with HFFB technology, can estimate the structural response boundaries of tall buildings and give nonstationary reference wind speeds. This method requires only conventional wind tunnel tests which are carried out in a stationary wind field.


### 2.2.1. Under Deterministic Excitation

As shown in Figure1, consider a simple single degree of freedom (SDOF) oscillator model, with mass m, spring stiffness k, damping coefficient c, and external force Ft. The motion equation can be given in the form as follows:(5)mz¨+cz˙+kz=Ft,or(6)z¨+2ζωnz˙+ωnz=Ftm,where z, z˙, and z¨ are the displacement, velocity, and acceleration response for the first mode in each direction of the system, respectively. The natural frequency ωn and damping ratio ζ are given by(7)ωn=km,cc=2mωn,ζ=ccc,where cc is the critical damping coefficient.Figure 1
SDOF spring-mass-damper vibration system.
(a)
(b)In this paper, the response of Equations (5) or (6) is solved using Duhamel’s integral. Herein, the piecewise linear interpolation function [24] is first used to approximately fit the sequence of Fti (denoted as Fi), where the rate of change of Ft within arbitrary period of time is supposed to be linear. In this way, Duhamel’s integral can be then applied to compute the response at time t=tj, j=1,2,3,…,n:(8)zj=ΔFjkΔtjΔtj−2ζωn+e−ζωnΔtj2ζωncosωdΔtj−ωd2−ζ2ωnωn2ωdsinωdΔtj+Fj−1k1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj+e−ζωnΔtjzj−1cosωdΔtj+z˙j−1+ζωnzj−1ωdsinωdΔtj,where ΔFj=Fj−Fj−1, Δtj=tj−tj−1, and ωd=ωn1−ζ2. The velocity z˙j at time t=tj can be calculated as follows:(9)z˙j=ΔFjkΔtj1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj+Fj−1ke−ζωnΔtjωn2ωdsinωdΔtj+e−ζωnΔtjz˙j−1cosωdΔtj−ζωnωdz˙j−1+ωnζzj−1sinωdΔtj.Equations (8) and (9) can be rewritten in the following form:(10)zj=AjΔFj+BjFj−1+Cjzj−1+Djz˙j−1,z˙j=A¯jΔFj+B¯jFj−1+C¯jzj−1+D¯jz˙j−1,where(11)Aj=1kΔtjΔtj−2ζωn+e−ζωnΔtj2ζωncosωdΔtj−ωd2−ζ2ωnωn2ωdsinωdΔtj,Bj=1k1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,Cj=e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,Dj=e−ζωnΔtj1ωdsinωdΔtj,A¯j=1kΔtj1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,B¯j=1ke−ζωnΔtjωn2ωdsinωdΔtj,C¯j=−e−ζωnΔtjωn2ωdsinωdΔtj,D¯j=e−ζωnΔtjcosωdΔtj−ζωnωdsinωdΔtj.Rewriting Equation (10) in matrix form gives(12)zjz˙j=AjBj−AjCjDjA¯jB¯j−A¯jC¯jD¯jFjFj−1zj−1z˙j−1.By using either Equations (10) or (12) repeatedly, the following recurrence sequences can be obtained:(13)FjFj−1zj−1z˙j−1=10000010000Aj−1Bj−1−Aj−1Cj−1Dj−10A¯j−1B¯j−1−A¯j−1C¯j−1D¯j−1FjFj−1Fj−2zj−2z˙j−2=Hj−1FjFj−1Fj−2zj−2z˙j−2,FjFj−1Fj−2zj−2z˙j−2=10000001000000100000Aj−2Bj−2−Aj−2Cj−2Dj−200A¯j−2B¯j−2−A¯j−2C¯j−2D¯j−2FjFj−1Fj−2Fj−3zj−3z˙j−3=Hj−2FjFj−1Fj−2Fj−3zj−3z˙j−3,FjFj−1⋮F1z1z˙1=10⋯000001⋯0000⋮⋮⋱00000001000000A1B1−A1B1D1000A¯1B¯1−A¯1B¯1D¯1FjFj−1⋮F1F0z0z˙0=H1FjFj−1⋮F1F0z0z˙0.Hence, there is:(14)FjFj−1zj−1z˙j−1=Hj−1,Hj−2,…,H1FjFj−1⋮F1F0z0z˙0.LetH∗=AjBj−AjCjDjA¯jB¯j−A¯jC¯jD¯j, H=H∗∏k=1j−1Hk, F=FjFj−1⋯F1F0Τ, and z0=z0z˙0Τ, then Equation (12) can be expressed as(15)zjz˙j=HFz0,where H is a 2×j+3 constant coefficient matrix, F is the vector of force sequence, and z0 is the initial condition.


### 2.2.2. Under Uncertain Excitation

In nonrandom vibration analysis of the interval process model, the external excitation is treated as an interval process,FIt=FLt,FUt, and the autocorrelation function is rti,tj. FIt can be represented by an interval vector, FI=FjIFj−1I⋯F1IF0IT (Equation (16)). Therefore, Equation (15) can be rewritten by eliminating correlation between variables and transforming them into an independent interval processes F¯It:(16)zjz˙j=HFIz0.F
I
t is obtained through Cholesky decomposition, by eliminating the correlation between variables and transforming them into independent interval processes, F¯It, with a corresponding interval vector, F¯I. The following equation can be obtained [23]:(17)F¯I=L−1FI.The corresponding interval middle point vector isF¯C, for which the following equation can be obtained:(18)F¯C=L−1FC.The following equation can be obtained for the corresponding interval half width vectorF¯W:(19)F¯W=L−1FW,where L is a j+1×j+1 lower triangular matrix, L denotes the matrix whose entries are absolute values of corresponding ones in the matrix L.Through this treatment, Equation (16) can be rewritten as [23]:(20)zjz˙j=HLF¯Iz0=HFHZLF¯Iz0=HFLF¯I+HZz0,where HF and HZ are block matrices of H, with dimensions 2×j+1 and 2×2, respectively.DenoteδF¯W as(21)δF¯W=δjIF¯jW⋯δ1IF¯1Wδ0IF¯0WT,δiI=−1,1,i=0,1,…,j.LetF¯I=F¯C−F¯W,FC+F¯W. According to Equations (18)–(20), F¯I can be rewritten as(22)F¯I=F¯C+δF¯W.Substituting Equations (22), (18), and (19) in Equation (20) yields(23)zjz˙j=HFLF¯I+HZz0=HFLF¯C+HFLδF¯W+HZz0∈HFLF¯C+HZz0−HFLF¯W,HFLF¯C+HZz0+HFLF¯W=HFLL−1FC+HZz0−HFLL−1FW,HFLL−1FC+HZz0−HFLL−1FW=HFFC+HZz0−HFLL−1FW,HFFC+HZz0−HFLL−1FW.This implies that the upper and lower bounds of the displacement response at timet=tj are(24a)zjU=HF1FC+HZ1z0+HF1LL−1FW,(24b)zjL=HF1FC+HZ1z0−HF1LL−1FW.Similarly, the upper and lower bounds of velocity response at timet=tj can be obtained as(25a)z˙jU=HF2FC+HZ2z0+HF2LL−1FW,(25b)z˙jL=HF2FC+HZ2z0−HF2LL−1FW.Equation (5) can be rewritten as(26)z¨j=−cmz˙j−kmzj+Ftm.Also(27)z˙j=z˙j.Rewrite Equations (26) and (27) in matrix form:(28a)z˙jz¨j=01−km−cmzjz˙j+01mFt.Fromzj∈zjL,zjU=zjI,z˙j∈z˙jL,z˙jU=z˙jI,z¨j∈z¨jL,z¨jU=z¨jI,Ft∈FLt,FUt=FIt, where zjI, z˙jI and z¨jI are the interval displacement response, interval velocity response, and interval acceleration response, respectively.Equation (28a) can be rewritten as(28b)z˙jIz¨jI=01−km−cmzjIz˙jI+01mFIt.Equation (26) indicates that the upper and lower bounds of acceleration response can be obtained as(29a)z¨jL=−cmz˙jU−kmzjU+FLtm,(29b)z¨jU=cmz˙jL−kmzjL+FUtm,where FUt is the upper bound of excitation force and FLt is the lower bound of excitation force.


## 2.2.1. Under Deterministic Excitation

As shown in Figure1, consider a simple single degree of freedom (SDOF) oscillator model, with mass m, spring stiffness k, damping coefficient c, and external force Ft. The motion equation can be given in the form as follows:(5)mz¨+cz˙+kz=Ft,or(6)z¨+2ζωnz˙+ωnz=Ftm,where z, z˙, and z¨ are the displacement, velocity, and acceleration response for the first mode in each direction of the system, respectively. The natural frequency ωn and damping ratio ζ are given by(7)ωn=km,cc=2mωn,ζ=ccc,where cc is the critical damping coefficient.Figure 1
SDOF spring-mass-damper vibration system.
(a)
(b)In this paper, the response of Equations (5) or (6) is solved using Duhamel’s integral. Herein, the piecewise linear interpolation function [24] is first used to approximately fit the sequence of Fti (denoted as Fi), where the rate of change of Ft within arbitrary period of time is supposed to be linear. In this way, Duhamel’s integral can be then applied to compute the response at time t=tj, j=1,2,3,…,n:(8)zj=ΔFjkΔtjΔtj−2ζωn+e−ζωnΔtj2ζωncosωdΔtj−ωd2−ζ2ωnωn2ωdsinωdΔtj+Fj−1k1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj+e−ζωnΔtjzj−1cosωdΔtj+z˙j−1+ζωnzj−1ωdsinωdΔtj,where ΔFj=Fj−Fj−1, Δtj=tj−tj−1, and ωd=ωn1−ζ2. The velocity z˙j at time t=tj can be calculated as follows:(9)z˙j=ΔFjkΔtj1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj+Fj−1ke−ζωnΔtjωn2ωdsinωdΔtj+e−ζωnΔtjz˙j−1cosωdΔtj−ζωnωdz˙j−1+ωnζzj−1sinωdΔtj.Equations (8) and (9) can be rewritten in the following form:(10)zj=AjΔFj+BjFj−1+Cjzj−1+Djz˙j−1,z˙j=A¯jΔFj+B¯jFj−1+C¯jzj−1+D¯jz˙j−1,where(11)Aj=1kΔtjΔtj−2ζωn+e−ζωnΔtj2ζωncosωdΔtj−ωd2−ζ2ωnωn2ωdsinωdΔtj,Bj=1k1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,Cj=e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,Dj=e−ζωnΔtj1ωdsinωdΔtj,A¯j=1kΔtj1−e−ζωnΔtjcosωdΔtj+ζωnωdsinωdΔtj,B¯j=1ke−ζωnΔtjωn2ωdsinωdΔtj,C¯j=−e−ζωnΔtjωn2ωdsinωdΔtj,D¯j=e−ζωnΔtjcosωdΔtj−ζωnωdsinωdΔtj.Rewriting Equation (10) in matrix form gives(12)zjz˙j=AjBj−AjCjDjA¯jB¯j−A¯jC¯jD¯jFjFj−1zj−1z˙j−1.By using either Equations (10) or (12) repeatedly, the following recurrence sequences can be obtained:(13)FjFj−1zj−1z˙j−1=10000010000Aj−1Bj−1−Aj−1Cj−1Dj−10A¯j−1B¯j−1−A¯j−1C¯j−1D¯j−1FjFj−1Fj−2zj−2z˙j−2=Hj−1FjFj−1Fj−2zj−2z˙j−2,FjFj−1Fj−2zj−2z˙j−2=10000001000000100000Aj−2Bj−2−Aj−2Cj−2Dj−200A¯j−2B¯j−2−A¯j−2C¯j−2D¯j−2FjFj−1Fj−2Fj−3zj−3z˙j−3=Hj−2FjFj−1Fj−2Fj−3zj−3z˙j−3,FjFj−1⋮F1z1z˙1=10⋯000001⋯0000⋮⋮⋱00000001000000A1B1−A1B1D1000A¯1B¯1−A¯1B¯1D¯1FjFj−1⋮F1F0z0z˙0=H1FjFj−1⋮F1F0z0z˙0.Hence, there is:(14)FjFj−1zj−1z˙j−1=Hj−1,Hj−2,…,H1FjFj−1⋮F1F0z0z˙0.LetH∗=AjBj−AjCjDjA¯jB¯j−A¯jC¯jD¯j, H=H∗∏k=1j−1Hk, F=FjFj−1⋯F1F0Τ, and z0=z0z˙0Τ, then Equation (12) can be expressed as(15)zjz˙j=HFz0,where H is a 2×j+3 constant coefficient matrix, F is the vector of force sequence, and z0 is the initial condition.


## 2.2.2. Under Uncertain Excitation

In nonrandom vibration analysis of the interval process model, the external excitation is treated as an interval process,FIt=FLt,FUt, and the autocorrelation function is rti,tj. FIt can be represented by an interval vector, FI=FjIFj−1I⋯F1IF0IT (Equation (16)). Therefore, Equation (15) can be rewritten by eliminating correlation between variables and transforming them into an independent interval processes F¯It:(16)zjz˙j=HFIz0.F
I
t is obtained through Cholesky decomposition, by eliminating the correlation between variables and transforming them into independent interval processes, F¯It, with a corresponding interval vector, F¯I. The following equation can be obtained [23]:(17)F¯I=L−1FI.The corresponding interval middle point vector isF¯C, for which the following equation can be obtained:(18)F¯C=L−1FC.The following equation can be obtained for the corresponding interval half width vectorF¯W:(19)F¯W=L−1FW,where L is a j+1×j+1 lower triangular matrix, L denotes the matrix whose entries are absolute values of corresponding ones in the matrix L.Through this treatment, Equation (16) can be rewritten as [23]:(20)zjz˙j=HLF¯Iz0=HFHZLF¯Iz0=HFLF¯I+HZz0,where HF and HZ are block matrices of H, with dimensions 2×j+1 and 2×2, respectively.DenoteδF¯W as(21)δF¯W=δjIF¯jW⋯δ1IF¯1Wδ0IF¯0WT,δiI=−1,1,i=0,1,…,j.LetF¯I=F¯C−F¯W,FC+F¯W. According to Equations (18)–(20), F¯I can be rewritten as(22)F¯I=F¯C+δF¯W.Substituting Equations (22), (18), and (19) in Equation (20) yields(23)zjz˙j=HFLF¯I+HZz0=HFLF¯C+HFLδF¯W+HZz0∈HFLF¯C+HZz0−HFLF¯W,HFLF¯C+HZz0+HFLF¯W=HFLL−1FC+HZz0−HFLL−1FW,HFLL−1FC+HZz0−HFLL−1FW=HFFC+HZz0−HFLL−1FW,HFFC+HZz0−HFLL−1FW.This implies that the upper and lower bounds of the displacement response at timet=tj are(24a)zjU=HF1FC+HZ1z0+HF1LL−1FW,(24b)zjL=HF1FC+HZ1z0−HF1LL−1FW.Similarly, the upper and lower bounds of velocity response at timet=tj can be obtained as(25a)z˙jU=HF2FC+HZ2z0+HF2LL−1FW,(25b)z˙jL=HF2FC+HZ2z0−HF2LL−1FW.Equation (5) can be rewritten as(26)z¨j=−cmz˙j−kmzj+Ftm.Also(27)z˙j=z˙j.Rewrite Equations (26) and (27) in matrix form:(28a)z˙jz¨j=01−km−cmzjz˙j+01mFt.Fromzj∈zjL,zjU=zjI,z˙j∈z˙jL,z˙jU=z˙jI,z¨j∈z¨jL,z¨jU=z¨jI,Ft∈FLt,FUt=FIt, where zjI, z˙jI and z¨jI are the interval displacement response, interval velocity response, and interval acceleration response, respectively.Equation (28a) can be rewritten as(28b)z˙jIz¨jI=01−km−cmzjIz˙jI+01mFIt.Equation (26) indicates that the upper and lower bounds of acceleration response can be obtained as(29a)z¨jL=−cmz˙jU−kmzjU+FLtm,(29b)z¨jU=cmz˙jL−kmzjL+FUtm,where FUt is the upper bound of excitation force and FLt is the lower bound of excitation force.


## 3. Data Process Method for HFFB Wind Tunnel Test

In the wind tunnel test, the mean wind speed profile is adjusted to match the target wind profile as(30)v=vz10zα,where vz is the mean wind speed at height z and α is the terrain roughness exponent.A wind tunnel test is a kind of experiment that uses reduced-scale models to represent the geometrical characteristics of a real building. Based on the principle of similitude, the measured wind forces and moments at the base of the building model can be transformed to the actual wind force acting on the building prototype. To establish the time, geometry, and wind speed similarity relationship between the model and the prototype, time scale ratioλT, geometrical scale ratio λL, and wind speed scale ratio λV are defined as Equations (31a)–(31c), respectively:(31a)λT=TmTfull,(31b)λL=LmLfull,(31c)λV=VmVfull,where Tm, Lm, and Vm are the time, geometrical size, and wind speed for the model, respectively. Tfull, Lfull, and Vfull are time, geometrical size, and wind speed for actual buildings, respectively.According to the principle of similitude, the above three satisfies(32)TmTfull=L/VmL/Vfull=Lm/LfullVm/Vfull=λLλV.SinceT=1/f, the scale ratio T of frequency can be obtained as(33)fmffull=1Tm/Tfull=1λT,where fm and ffull are the frequencies for the model and actual building, respectively.In conventional data analysis of a wind tunnel test, time or frequency scale is a constant for a certain wind direction; however, if a real wind field is nonstationary, the time scale is time-variant. In this study, the entire considered time is separated into a series of time segments defined asΔti, which can be formulated as(34)Δti=1ffull=VmLfullfmLm1Vfulli,i=1,2,…,n.Conventionally, aerodynamic force and moment coefficients are employed to implement the model-to-prototype transform. The torque coefficientsμx, μy, and μz are defined as(35a)μx=mx0.5ρVh2BmHm2,(35b)μy=my0.5ρVh2BmHm2,(35c)μz=mz0.5ρVh2Bm2Hm,where mx, my, and mz are the base bending moments measured in the direction x, y, and z, respectively, ρ is the air density, Vh is the mean wind speed at the model reference height, Bm is the model characteristic width, and Hm is the total height of the model. Through the torque coefficients μx, μy, and μz, the lower and upper bounds of the base moment of the real building can be formulated as(36a)MLx=0.5ρVL2BfullHfull2μx,MUx=0.5ρVU2BfullHfull2μx,(36b)MLy=0.5ρVL2BfullHfull2μy,MUy=0.5ρVU2BfullHfull2μy,(36c)MLz=0.5ρVL2Bfull2Hfullμz,MUz=0.5ρVU2Bfull2Hfullμz,where MUx, MUy, and MUz are the upper bounds of the base bending moments of the actual structure in x, y, and z direction, respectively; MLx, MLy, and MLz are the corresponding lower bounds; VU and VL are the upper and lower bounds of the time-variant wind speed; and Bfull and Hfull are the characteristic width and total height of the real buildings, respectively.A high-rise building can be examined using a cantilever structure response analysis which satisfies the motional equation as(37)mz∂2y∂t2+cz∂y∂t+∂2∂z2EIz∂2y∂z2=pz,t,where mz, cz, and EIz are the mass, damping coefficient, and flexural rigidity per unit length at height z, respectively, and pz,t is the horizontal wind load at height z and time instant t.The modal massmp, modal stiffness kp, and modal damping coefficient cp for the first mode can be computed as follows:(38)mp=∫0Hmzφ2zdz=1H2∫0Hz2mzdz,kp=1H2∫0HEIzz2dz=ω02mp,cp=2ζω0mp.Then, the structural dynamic equation in modal coordinates can be formulated as(39)mpq¨+cpq˙+kpq=Ft.Assuming the modal shape for the first two sway modes to be linear with height, modal force can be expressed as(40)Ft=∫0Hφzpz,tdz=1H∫0Hzpz,tdz=MItH,where MIt is the base bending moment, which can be measured by the HFFB wind tunnel test.The whole procedure of the proposed method for estimation of boundary response of high-rise buildings subjected to nonstationary wind load is listed below:(1)
Get the upper and lower bounds of the reference wind speed conventionally from meteorological agencies.(2)
Measure the base forces and torques through a wind tunnel test.(3)
Using the base overturning moments, compute the base overturning moment coefficient according to Equations (35a)–(35c).(4)
Using Equations (36a)–(36c), compute the upper and lower bounds of structural modal force by the following equations:(41a)FxL=0.5ρVL2BfullHfull2μxHfull,FxU=0.5ρVU2BfullHfull2μxHfull,(41b)FyL=0.5ρVL2BfullHfull2μyHfull,FyU=0.5ρVU2BfullHfull2μyHfull,(41c)FzL=0.5ρVL2Bfull2HfullμzHfull,FzU=0.5ρVU2Bfull2HfullμzHfull,where FxU, FyU, and FzU are the upper bounds of the modal force of the actual structure in the x, y, and z directions, respectively, and FxL, FyL, and FzL are the corresponding lower bounds.(5)
Compute upper and lower bounds of structural response using Equations (24a), (24b), (25a), (25b), (29a), and (29b).


## 4. Case Study

In this paper, the feasibility of the interval process model and nonrandom vibration analysis method [23] in practical engineering is explained based on dynamic excitation as a stationary interval process, dynamic excitation as a nonstationary interval process, and the engineering example of GZET.


### 4.1.FIt Stationary

As shown in Figure1, consider a spring-mass-damping system where the mass m=1, damping coefficient c=0.2, and spring stiffness k=1. Herein, Ft is treated as a stationary interval process FIt, Ft∈10,20, where the middle point function is Fct=15 and half width function is Frt=5. The initial condition is z0=z˙0=0. The autocorrelation coefficient function is rτ.The autocorrelation coefficient function is assumed to take the form ofrτ=e−λτ. Figures 2–4 show the boundary response of displacement, velocity, and acceleration for the case of λ=0.2. We find that the response bounds of displacement can be roughly divided into two phases: the transient response stage and the steady-state response stage. Before 30 s, the response bounds exhibit obvious oscillation. This kind of oscillation tends to reduce with time until a steady state can be reached. Figures 5–7 show the stationary interval process autocorrelation coefficient function rτ=e−λτ. Cases with different λ values are considered to observe the effect of λ on the obtained boundary response. As can be seen from the results, as λ increases, the interval of the system’s steady response phase will increase, indicating that the self-correlation function of the time-varying force, FIt, may have a significant impact on dynamic responses, such as displacement and velocity, when the incentive interval is not changed.Figure 2
Displacement response bounds of a stationary interval process.Figure 3
Velocity response bounds of a stationary interval process.Figure 4
Acceleration response bounds of a stationary interval process.Figure 5
Dynamic displacement response under different values of parameterλ.Figure 6
Dynamic velocity response under different values of parameterλ.Figure 7
Dynamic acceleration response under different values of parameterλ.


### 4.2.FIt Is Nonstationary

This case also focuses on the damped SDOF system in Figure1, and the values of relevant parameters are the same as those in Section 4.1. The difference is FIt. It is a nonstationary interval process, as shown in Figure 8, where the middle point function is Fct=16+2sint and half width function is Frt=2+sint. Figures 9–11 show the displacement response, velocity response, and acceleration response of the nonstationary interval process. We find that because FIt exhibits a variation of periodicity, the response also exhibits a variation of periodicity, and the periodicity changes are very similar.Figure 8
The nonstationary excitation forceFIt.Figure 9
Displacement response bounds of a nonstationary interval process.Figure 10
Velocity response bounds of a nonstationary interval process.Figure 11
Acceleration response bounds of a nonstationary interval process.Figures12–14 show the stationary interval process autocorrelation coefficient function rτ=e−λτ. Cases with different λ values are considered to observe the effect of λ on the obtained boundary response. As can be seen from the results, as λ increases, the interval of the system’s steady response phase will increase, indicating that the self-correlation function of the time-varying force, FIt, may have a significant impact on dynamic responses, such as displacement and velocity, when the incentive interval is not changed.Figure 12
Dynamic displacement response under different values of parameterλ.Figure 13
Dynamic velocity response under different values of parameterλ.Figure 14
Dynamic acceleration response under different values of parameterλ.


### 4.3. The GZET Case Study

In this section, a supertall building, GZET, is used as an example to show the effectiveness of the proposed method. GZET, which is 530 m high with 112 stories, is in Guangzhou, China. Currently it is the tallest building in Guangzhou and the second tallest building in South China.An HFFB wind tunnel test of GZET was carried out in the wind tunnel laboratory of South China University of Technology. The experimental section of the wind tunnel laboratory is 5 m wide, 3 m high, and 24 m long. The wind speed of the test section can be adjusted continuously, and the maximum wind speed can reach 30 m/s. A rigid model with a geometric length scale of 1 : 500 was made to represent GZET. The model was made of balsa and foam and was mounted on a metal slab which was rigidly connected to an HFFB measurement system. Models of surrounding buildings that might affect the wind’s effects on the tested building were also made and mounted on the ground of the wind tunnel to simulate the surrounding conditions, as shown in Figure15. A Pitot tube was installed at the height of the building model (1.06 m above ground) to measure the reference wind speed. The wind direction was defined as an angle, β, from the east along an anticlockwise direction, varying from 0° to 360° with an increment angle of 10°. Before the formal HFFB wind tunnel test was carried out, the wind profile in the wind tunnel had been adjusted to match the ESDU’s wind profile with the exponential parameter α=0.22, as shown in Figure 16. In the wind tunnel test, the sampling frequency is 400 Hz with a sampling length of 40960. The natural frequency of GZET is 0.128 and 0.132 Hz for the first two sway modes and 0.360 Hz for the first torsional mode. According to the actual engineering situation, the correlation coefficient function of wind load is obtained directly from wind tunnel test to compute the response boudary of the structure.Figure 15
Building model of the GZET in the wind tunnel.Figure 16
Mean wind speed and turbulence intensity profile. (a) Mean wind speed; (b) turbulence intensity.
(a)
(b)In the wind-induced vibration response analysis, using the wind speed of a certain typhoon provided by the meteorological department, 3,600 samples were taken to measure the typhoon wind speed, as shown in Figure17, and the upper and lower envelope diagrams thereof were plotted according to Figure 17, as shown in Figure 18. Using the upper and lower boundaries of the typhoon wind speed combined with the HFFB wind tunnel test data, structural response boundary values can be obtained. Structural response boundary values are shown in Figures 19–21, which are the displacement, velocity, and acceleration responses of the structure, respectively. To better observe the changing trends of response at each time, the envelopes of structure displacement, velocity, and acceleration responses were, respectively, fitted. In Figures 19–21, the blue curve is the structural boundary response obtained by the interval analysis method, and the red curve is the envelope of the boundary response method which is more intuitive for applications. The envelope diagram can reflect the response at each moment more intuitively and clearly.Figure 17
Segment of field-measured wind speed.Figure 18
Envelope of the wind speed.Figure 19
Boundary response of displacement atop the building.Figure 20
Boundary response of velocity atop the building.Figure 21
Boundary response of acceleration atop the building.


## 4.1.FIt Stationary

As shown in Figure1, consider a spring-mass-damping system where the mass m=1, damping coefficient c=0.2, and spring stiffness k=1. Herein, Ft is treated as a stationary interval process FIt, Ft∈10,20, where the middle point function is Fct=15 and half width function is Frt=5. The initial condition is z0=z˙0=0. The autocorrelation coefficient function is rτ.The autocorrelation coefficient function is assumed to take the form ofrτ=e−λτ. Figures 2–4 show the boundary response of displacement, velocity, and acceleration for the case of λ=0.2. We find that the response bounds of displacement can be roughly divided into two phases: the transient response stage and the steady-state response stage. Before 30 s, the response bounds exhibit obvious oscillation. This kind of oscillation tends to reduce with time until a steady state can be reached. Figures 5–7 show the stationary interval process autocorrelation coefficient function rτ=e−λτ. Cases with different λ values are considered to observe the effect of λ on the obtained boundary response. As can be seen from the results, as λ increases, the interval of the system’s steady response phase will increase, indicating that the self-correlation function of the time-varying force, FIt, may have a significant impact on dynamic responses, such as displacement and velocity, when the incentive interval is not changed.Figure 2
Displacement response bounds of a stationary interval process.Figure 3
Velocity response bounds of a stationary interval process.Figure 4
Acceleration response bounds of a stationary interval process.Figure 5
Dynamic displacement response under different values of parameterλ.Figure 6
Dynamic velocity response under different values of parameterλ.Figure 7
Dynamic acceleration response under different values of parameterλ.


## 4.2.FIt Is Nonstationary

This case also focuses on the damped SDOF system in Figure1, and the values of relevant parameters are the same as those in Section 4.1. The difference is FIt. It is a nonstationary interval process, as shown in Figure 8, where the middle point function is Fct=16+2sint and half width function is Frt=2+sint. Figures 9–11 show the displacement response, velocity response, and acceleration response of the nonstationary interval process. We find that because FIt exhibits a variation of periodicity, the response also exhibits a variation of periodicity, and the periodicity changes are very similar.Figure 8
The nonstationary excitation forceFIt.Figure 9
Displacement response bounds of a nonstationary interval process.Figure 10
Velocity response bounds of a nonstationary interval process.Figure 11
Acceleration response bounds of a nonstationary interval process.Figures12–14 show the stationary interval process autocorrelation coefficient function rτ=e−λτ. Cases with different λ values are considered to observe the effect of λ on the obtained boundary response. As can be seen from the results, as λ increases, the interval of the system’s steady response phase will increase, indicating that the self-correlation function of the time-varying force, FIt, may have a significant impact on dynamic responses, such as displacement and velocity, when the incentive interval is not changed.Figure 12
Dynamic displacement response under different values of parameterλ.Figure 13
Dynamic velocity response under different values of parameterλ.Figure 14
Dynamic acceleration response under different values of parameterλ.


## 4.3. The GZET Case Study

In this section, a supertall building, GZET, is used as an example to show the effectiveness of the proposed method. GZET, which is 530 m high with 112 stories, is in Guangzhou, China. Currently it is the tallest building in Guangzhou and the second tallest building in South China.An HFFB wind tunnel test of GZET was carried out in the wind tunnel laboratory of South China University of Technology. The experimental section of the wind tunnel laboratory is 5 m wide, 3 m high, and 24 m long. The wind speed of the test section can be adjusted continuously, and the maximum wind speed can reach 30 m/s. A rigid model with a geometric length scale of 1 : 500 was made to represent GZET. The model was made of balsa and foam and was mounted on a metal slab which was rigidly connected to an HFFB measurement system. Models of surrounding buildings that might affect the wind’s effects on the tested building were also made and mounted on the ground of the wind tunnel to simulate the surrounding conditions, as shown in Figure15. A Pitot tube was installed at the height of the building model (1.06 m above ground) to measure the reference wind speed. The wind direction was defined as an angle, β, from the east along an anticlockwise direction, varying from 0° to 360° with an increment angle of 10°. Before the formal HFFB wind tunnel test was carried out, the wind profile in the wind tunnel had been adjusted to match the ESDU’s wind profile with the exponential parameter α=0.22, as shown in Figure 16. In the wind tunnel test, the sampling frequency is 400 Hz with a sampling length of 40960. The natural frequency of GZET is 0.128 and 0.132 Hz for the first two sway modes and 0.360 Hz for the first torsional mode. According to the actual engineering situation, the correlation coefficient function of wind load is obtained directly from wind tunnel test to compute the response boudary of the structure.Figure 15
Building model of the GZET in the wind tunnel.Figure 16
Mean wind speed and turbulence intensity profile. (a) Mean wind speed; (b) turbulence intensity.
(a)
(b)In the wind-induced vibration response analysis, using the wind speed of a certain typhoon provided by the meteorological department, 3,600 samples were taken to measure the typhoon wind speed, as shown in Figure17, and the upper and lower envelope diagrams thereof were plotted according to Figure 17, as shown in Figure 18. Using the upper and lower boundaries of the typhoon wind speed combined with the HFFB wind tunnel test data, structural response boundary values can be obtained. Structural response boundary values are shown in Figures 19–21, which are the displacement, velocity, and acceleration responses of the structure, respectively. To better observe the changing trends of response at each time, the envelopes of structure displacement, velocity, and acceleration responses were, respectively, fitted. In Figures 19–21, the blue curve is the structural boundary response obtained by the interval analysis method, and the red curve is the envelope of the boundary response method which is more intuitive for applications. The envelope diagram can reflect the response at each moment more intuitively and clearly.Figure 17
Segment of field-measured wind speed.Figure 18
Envelope of the wind speed.Figure 19
Boundary response of displacement atop the building.Figure 20
Boundary response of velocity atop the building.Figure 21
Boundary response of acceleration atop the building.


## 5. Conclusions

Based on the HFFB technique and interval process model analysis theory, this paper proposes a new method for estimating the response boundary of high-rise building structures based on nonstationary wind loads. With a given nonstationary reference wind speed, this method can provide the upper and lower bounds for displacement, velocity, and acceleration response for different time instants. With the predicted wind information of a typhoon, it is easy to know the possible largest response for different time segments, which helps the owner of a building take necessary measures to prevent a typhoon disaster and reduce damage. This new method will help researchers and engineers prevent and control typhoon disasters in high-rise buildings.



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*Source: 3275302-2018-10-01.xml*