Wind-Induced Dynamic Response of Inter-Story Isolated Tall Buildings with Friction Pendulum Bearing Based on an Enhanced Simplified Model

Abstract

Isolation technology, especially for base isolation, is increasingly being applied in earthquake-prone areas. To satisfy some special demands (such as prevention from seawater erosion of an isolation layer, story-adding retrofit of existing buildings, avoidance of collision between base-isolated tall buildings, and so on), the isolation layer sometimes has to be set in the middle of a building to constitute inter-story isolated buildings. This special structural form inevitably encounters strong wind loads during service life, and its wind-resistant performance deserves to be investigated. This study conducts the wind-induced vibration analysis of inter-story isolated tall buildings with friction pendulum bearing (FPB). The nonlinear time domain analysis model and statistical linearization method to compute the wind-induced response of FPB inter-story isolated tall buildings are addressed based on an enhanced simplified model. Considering the independence of the upper and lower structures, two structural design schemes for inter-story isolated tall buildings are provided. Their dynamic characteristics are analyzed, and wind-induced responses are compared. Finally, the accuracy of the statistical linearization method is verified. This study provides an important theoretical basis for the structural design and wind resistance of inter-story isolated tall buildings.

1. Introduction

As one of the most effective earthquake-resistant technologies, base isolation has demonstrated its superior seismic performance after more than half a century of theoretical development and practical application [1,2,3]. However, in engineering practice, the isolation layer sometimes has to be installed on the higher floor to prevent the seawater erosion of buildings located in coastal areas, or reduce excessive isolation layer displacement of those in high-density building areas [4]. Additionally, the isolation system is adopted to serve as a “soft transfer” between the rigid chassis and flexible towers in order to avoid the stiffness discontinuity. Furthermore, the isolation technology is applied in retrofit strategy on existing buildings [5,6,7]. These buildings are divided into two parts by the isolation layer to realize multi-functionality, such as commercial and residential functions [8]. Thus, it has gained more attention and rapid development. Japan is at the forefront of the world in the field of building isolation and has built a few inter-story isolated tall buildings [8]. The highest one, Roppongi Grand Tower, is up to 230 m [9].

Over the past two decades, extensive theoretical analyses have been conducted on the seismic resistance of inter-story isolated buildings [10,11,12,13,14,15]. Recently, some research based on engineering cases has also been carried out, including shaking table tests on over-track multi-tower building [16,17] and story-adding retrofit of existing buildings by isolation bearings [18]. The structural design of an inter-story isolated building is complex due to more dynamic parameters being involved. By specifying optimal control objectives, single- or multi-objective optimization methods based on genetic algorithms are used to determine these dynamic parameters [19,20,21,22].

The above study on the seismic performance of inter-story isolated buildings has obtained enough attention, and these efforts have laid the foundation for practical application of this type of structure. However, taller inter-story isolated buildings inevitably encounter strong wind loads during service life. The mainstream research of classical wind engineering is the wind-induced vibrations of tall buildings and long-span bridges [23,24,25]. The research on the wind resistance of base-isolated tall buildings has become a hot topic in recent years. Siringoringo and Fujino [26] performed the real-time monitoring of a base-isolated building subject to typhoon. Feng and Chen [27,28] systematically studied the nonlinear alongwind and crosswind responses of base-isolated tall buildings with lead rubber bearings. Li et al. [29] provided the equivalent static wind load of the base-isolated tall building from the perspective of the load specification. Li et al. [30,31] investigated the uncoupled and three-dimensional coupled wind-induced response of a base-isolated tall building with friction pendulum bearing (FPB). Nevertheless, there is almost no report on the wind resistance of inter-story isolated tall buildings.

As the more generalized version of the base-isolated building, inter-story isolated tall buildings consist of three substructures: the lower structure (LS), isolation layer, and upper structure (US). The dynamic parameters of substructures have significant influences on the dynamic responses of those three parts. An efficient and accurate evaluation of the wind-induced response for this unique structural form is significant. In fact, the computation of wind-induced vibration of an isolated tall building is time-consuming, as the nonlinear mechanical characteristics of the isolation layer, multiple degrees of freedom (DOFs) involved in the dynamic model, and a large scale of sample of wind load time history are necessary to obtain the response statistic. The nonlinear time domain analysis of the structural system with a large number of DOFs could be inefficient for optimization problems in which the massive iterative computation is involved. Seeking an accurate and reasonable simplified model, and developing an effective statistical linearization method are important to improve the efficiency of nonlinear wind vibration analysis.

This paper extends the previous works [30,31] on the wind resistance of the FPB base-isolated tall building to the FPB inter-story isolated tall building. An analytical framework of the wind-induced vibration of an FPB inter-story isolated tall building by the nonlinear time domain analysis and statistical linearization method is first proposed. To improve computational efficiency, a reasonable simplified model with a few DOFs of the LS and US is used in the time domain analysis and statistical linearization method. Secondly, considering the independence of the LS and US, two structural design schemes for the inter-story isolated building are suggested, their modal characteristics are analyzed, and crosswind responses are compared. Then, the accuracy of the statistical linearization method is verified. Some important conclusions are highlighted at the end of the paper.

2. Theoretical Framework

2.1. Equations of Motion

As shown in Figure 1, an inter-story isolated building with friction pendulum bearing (FPB) consists of an M-story LS, isolation layer, and N-story US. The mass, damping, stiffness, and external load on the i-th story of the LS are $m_{l,i}$, $c_{l,i}$, $k_{l,i}$, and $P_{l,i}\left(t\right)$ (i = 1, 2, …, M), respectively. The counterparts on the j-th story of the US are denoted as $m_{u,j}$, $c_{u,j},$ $k_{u,j}$, and $P_{u,j}\left(t\right)$ (j = 1, 2, …, N), respectively. For the isolation layer, its mass, damping, and external load are $m_b$, $c_b$, and $P_b\left(t\right)$, respectively. The soil–structure interaction and dynamic interaction between adjacent structures are neglected in this study.

2.1.1. Physical Coordinates

The displacement of the isolation layer, with respect to the top story of the LS, is denoted as $u_b$. The displacement vectors of the LS and US are represented as $Y_l={[y_{l,1},y_{l,2},\dots ,y_{l,M}]}^{\mathrm{T}}$ and $Y_u={[y_{u,1},y_{u,2},\dots ,y_{u,N}]}^{\mathrm{T}}$, respectively, where $Y_u$ is relative to the isolation slab. In physical coordinates, the equations of motion corresponding to three substructures of the inter-story isolated building are given by

$$M_l{\ddot{Y}}_l+C_l{\dot{Y}}_l+K_l{Y}_l+F_{l,\mathrm{inertial}}=P_l+F_{l,\mathrm{load}}$$

(1a)

$$\left(r_N^{\mathrm{T}}{M}_u{r}_N+m_b\right)\left({\ddot{y}}_{l,M}+{\ddot{u}}_b\right)+c_b{\dot{u}}_b+\left(W/R\right)u_b+F_f+r_N^{\mathrm{T}}{M}_u{\ddot{Y}}_u=p_b+r_N^{\mathrm{T}}{P}_u$$

(1b)

$$M_u{r}_N\left({\ddot{y}}_{l,M}+{\ddot{u}}_b\right)+M_u{\ddot{Y}}_u+C_u{\dot{Y}}_u+K_u{Y}_u=P_u$$

(1c)

$$F_{l,\mathrm{inertial}}=\left[\begin{array}{c}{\mathbf{0}}_{\left(M-1\right)\times 1}\\ m_b\left({\ddot{u}}_b+{\ddot{y}}_{l,M}\right)+r_N^{\mathrm{T}}{M}_u\left(r_N\left({\ddot{u}}_b+{\ddot{y}}_{l,M}\right)+{\ddot{Y}}_u\right)\end{array}\right]$$

(2a)

$$F_{l,\mathrm{load}}=\left[\begin{array}{c}{\mathbf{0}}_{\left(M-1\right)\times 1}\\ P_b\left(t\right)+r_N^{\mathrm{T}}{P}_u\end{array}\right]$$

(2b)

where $M_l$, $C_l$, and $K_l$ are the mass, damping, and stiffness matrices of the LS, respectively; $M_u$, $C_u$, and $K_u$ are the same metrices of the US; $P_l={[P_{l, 1}\left(t\right), P_{l, 2}\left(t\right), \dots , P_{l, M}\left(t\right)]}^T$ and $P_u={[P_{u, 1}\left(t\right), P_{u, 2}\left(t\right), \dots , P_{u, N}\left(t\right)]}^T$ are the wind load vectors along the stories of the LS and US, respectively; the force vectors $F_{l,\mathrm{inertial}}$ and $F_{l,\mathrm{load}}$ represent the inertial and external forces which act on the top story of the LS; $r_N$ is the N-dimension unit column vector; the total weight of the US and isolation slab is $W=mg$ where $m=r_N^T{M}_u{r}_N+m_b$ and $g=9.8 \mathrm{m}/{\mathrm{s}}^2$; $R$ is the radius of the sliding plate. $F_f$ represents the friction force of the bearings, which is expressed as [32]

$$F_f=\mu \left({\dot{u}}_b\right)W{z}_b$$

(3a)

$$\mu \left({\dot{u}}_b\right)={\mu }_{max}-\left({\mu }_{max}-{\mu }_{min}\right)\exp \left(-a\left|{\dot{u}}_b\right|\right)$$

(3b)

$${\dot{z}}_b=\frac{1}{Y}\left({\dot{u}}_b-\gamma \left|{\dot{u}}_b\right|{\left|z_b\right|}^{\eta -1}{z}_b-\beta {\dot{u}}_b{\left|z_b\right|}^{\eta }\right)$$

(3c)

where ${\mu }_{min}$ and ${\mu }_{max}$ are the friction coefficients when the velocity of isolation slab ${\dot{u}}_b=0$ and ${\dot{u}}_b=\infty$, respectively. Factor $a$ controls the variation of the friction coefficient from ${\mu }_{min}$ to ${\mu }_{max}$; and $z_b$ is the normalized hysteresis displacement. The parameters controlling the hysteresis loop shape need to meet the conditions $\gamma +\beta =1$ $(\gamma , \beta >0)$ and $\eta$ ≥ 1. A set of reasonable parameters $\gamma =\beta =0.5$, $\eta =2$ is used in this study. The low yield displacement is taken as $Y=0.13 \mathrm{mm}$.

In this study, the shear building model is used. The mass and stiffness matrices of the LS and US are represented as

$$M_l=\mathrm{diag}\left(m_{l,1},m_{l,2},\dots ,m_{l,M}\right);$$

(4a)

$$M_u=\mathrm{diag}\left(m_{u,1},m_{u,2},\dots ,m_{u,N}\right);$$

(4b)

$$K_l=\left[\begin{array}{l}{k}_{l,1}+k_{l,2}-k_{l,2} 0\\ k_{l,2}+k_{l,3}-k_{l,3} \\ \ddots \ddots \\ \mathrm{Sym} k_{l,M-1}+k_{l,M}-k_{l,M}\\ k_{l,M} \end{array}\right];$$

(4c)

$$K_u=\left[\begin{array}{l}{k}_{u,1}+k_{u,2}-k_{u,2} 0\\ k_{u,2}+k_{u,3}-k_{u,3} \\ \ddots \ddots \\ \mathrm{Sym} k_{u,N-1}+k_{u,N}-k_{u,N}\\ k_{u,N} \end{array}\right]$$

(4d)

Because the LS can be regarded as a fixed-base building, its damping matrix $C_l$ adopts Rayleigh damping [33], i.e., $C_l=a_{l0}{M}_l+a_{l1}{K}_l$, where coefficients $a_{l0}$ and $a_{l1}$ are determined by giving any two modal damping ratios. The US and isolation layer constitute a base-isolated building. A stiffness-proportional damping matrix is appropriate for the US, i.e., $C_u=a_{u1}{K}_u$ [34].

The cross-power spectral density of fluctuating wind loads on the i-th and j-th stories can be derived from the base bending moment according to the Japanese code [28,29]

$$S_{P_i{P}_j}\left(f\right)=S_{P_0}\left(f\right){(\frac{h_i}{H})}^{{\alpha }_i}(\frac{h_j}{H}{)}^{{\alpha }_i}\mathit{exp}(-\frac{k_{y}fH}{U_H}\frac{\left|h_i-h_j\right|}{H})$$

(5a)

$$S_{R_0}\left(f\right)={(\frac{1}{2}\rho U_H^{2}B{H}_0)}^2{S}_{C_M}\left(f\right)/{\left|J_y\left(f\right)\right|}^2$$

(5b)

$${\left|J_y\left(f\right)\right|}^2={{\displaystyle \int }}_0^H{{\displaystyle \int }}_0^H{(\frac{Z_i}{H})}^{1+{\alpha }_s}(\frac{Z_j}{H}{)}^{1+{\alpha }_s}\mathit{exp}(-\frac{k_{y}fH}{U_H}\frac{\left|z_i-z_j\right|}{H})\frac{1}{H}\frac{1}{H}{\mathrm{dz}}_i{\mathrm{dz}}_j$$

(5c)

where $h_i$ is the distance between the i-th story and ground; $H$ and $B$ are the height and width of the building; $H_0$ is the height of each story; ${\alpha }_s$ is the wind profile coefficient; $\rho$ is the air density; $k_y$ is the decay factor; $U_H$ is the mean wind speed at the building top; and $S_{c_M}\left(f\right)$ is the power spectral density (PSD) of the coefficient of the base bending moment [35]. Based on the PSD matrix of the story forces, the time history of fluctuating wind load is generated stochastically by the spectral representation method [36]. When the load on stories is non-stationary wind, the corresponding non-stationary wind-induced responses are also obtained [37].

2.1.2. Modal Coordinates

The displacements of the LS and US can be expressed, respectively, by their first r- and s-order modes (r $\le$ M; s $\le$ N):

$$\left\{\begin{array}{l}{Y}_l={\Phi }_l{q}_l\\ Y_u={\Phi }_u{q}_u\end{array}\right.$$

(6)

where ${\Phi }_l=\left[\begin{array}{cccc}{\mathbf{\Phi }}_{ l1}& {\mathbf{\Phi }}_{ l2}& \cdots & {\mathbf{\Phi }}_{ lr}\end{array}\right]$ and ${\Phi }_u=\left[\begin{array}{cccc}{\mathbf{\Phi }}_{ u1}& {\mathbf{\Phi }}_{ u2}& \cdots & {\mathbf{\Phi }}_{ u\mathrm{s}}\end{array}\right]$ are the modal matrix composed of the first r-order mode shape of the LS and the first s-order mode shape of the US, respectively. They can be determined by the eigenvalue analysis using mass and stiffness matrices. For convenience, the last elements ${\varphi }_{ li,M}$ of ${\mathbf{\varphi }}_{ li}$ (i = 1, 2, …, r) and ${\varphi }_{ uj,N}$ of ${\mathbf{\varphi }}_{ uj}$ (j = 1, 2, …, s) are normalized as the unit.

Substituting $Y_l$ and $Y_u$ in Equation (6) into Equations (1a)–(1c), and then premultiplying ${\Phi }_l^{\mathrm{T}}$ and ${\Phi }_u^{\mathrm{T}}$ on both sides of Equation (1a) and Equation (1c), respectively, the equations of motion with dimensionless coefficients are derived as

$$\left(r_{li}+1\right){\ddot{q}}_{li}+{{\displaystyle \sum }}_{k\ne i}^r{\ddot{q}}_{lk}+2{\omega }_{li}{\zeta }_{li}{r}_{li}{\dot{q}}_{li}+{\omega }_{li}^2{r}_{li}{q}_{li}+{\ddot{u}}_b+{{\displaystyle \sum }}_{j=1}^s{\Gamma }_{uj}{r}_{uj}{\ddot{q}}_{uj}=r_{li}{Q}_{li}\left(t\right)+Q_b\left(t\right)$$

(7a)

$${{\displaystyle \sum }}_{i=1}^r{\ddot{q}}_{li}+{\ddot{u}}_b+2{\omega }_b{\zeta }_b{\dot{u}}_b+{\omega }_b^2{u}_b+\mu \left({\dot{u}}_b\right)g{z}_b+{{\displaystyle \sum }}_{j=1}^s{\Gamma }_{uj}{r}_{uj}{\ddot{q}}_{uj}=Q_b\left(t\right)$$

(7b)

$${\Gamma }_{uj}{{\displaystyle \sum }}_{i=1}^r{\ddot{q}}_{li}+{\Gamma }_{uj}{\ddot{u}}_b+{\ddot{q}}_{uj}+2{\omega }_{uj}{\zeta }_{uj}{\dot{q}}_{uj}+{\omega }_{uj}^2{q}_{uj}=Q_{uj}\left(t\right)$$

(7c)

where $r_{li}=\frac{{\mathbf{\varphi }}_{ li}^T{M}_l{\mathbf{\varphi }}_{ li}}{m}$ and $r_{uj}=\frac{{\mathbf{\varphi }}_{ uj}^T{M}_u{\mathbf{\varphi }}_{ uj}}{m}$ are the mass ratios of the LS and US, respectively; ${\Gamma }_{li}=\frac{{\mathbf{\varphi }}_{ li}^T{M}_l{r}_M}{{\mathbf{\varphi }}_{ li}^T{M}_l{\mathbf{\varphi }}_{ li}}$ and ${\Gamma }_{uj}=\frac{{\mathbf{\varphi }}_{ uj}^T{M}_u{r}_N}{{\mathbf{\varphi }}_{ uj}^T{M}_u{\mathbf{\varphi }}_{ uj}}$ are the modal participation mass factors; ${\omega }_{li}=\sqrt{\frac{{\mathbf{\varphi }}_{ li}^T{K}_l{\mathbf{\varphi }}_{ li}}{{\mathbf{\varphi }}_{ li}^T{M}_l{\mathbf{\varphi }}_{ li}}}$ and ${\omega }_{uj}=\sqrt{\frac{{\mathbf{\varphi }}_{ uj}^T{K}_u{\mathbf{\varphi }}_{ uj}}{{\mathbf{\varphi }}_{ uj}^T{M}_u{\mathbf{\varphi }}_{ uj}}}$ are the modal frequencies of the LS and US, respectively; ${\zeta }_{li}=\frac{{\mathbf{\varphi }}_{ li}^T{C}_l{\mathbf{\varphi }}_{ li}}{2{\mathbf{\varphi }}_{ li}^T{M}_l{\mathbf{\varphi }}_{ li}{\omega }_{li}}$ and ${\zeta }_{uj}=\frac{{\mathbf{\varphi }}_{ uj}^T{C}_u{\mathbf{\varphi }}_{ uj}}{2{\mathbf{\varphi }}_{ uj}^T{M}_u{\mathbf{\varphi }}_{ uj}{\omega }_{uj}}$ are the modal damping ratios; ${\omega }_b=\sqrt{\frac{g}{R}}$ and ${\zeta }_b=\frac{c_b}{2m{\omega }_b}$ are the frequency and linear damping ratio of the isolation layer; and the corresponding inputs for three parts are $Q_{li}\left(t\right)=\frac{{\mathbf{\varphi }}_{ li}^T{P}_l}{{\mathbf{\varphi }}_{ li}^T{M}_l{\mathbf{\varphi }}_{ li}}$, $Q_b\left(t\right)=\frac{P_b+r_N^T{P}_u}{m}$, and $Q_{uj}\left(t\right)=\frac{{\mathbf{\varphi }}_{ uj}^T{P}_u}{{\mathbf{\varphi }}_{ uj}^T{M}_u{\mathbf{\varphi }}_{ uj}}$.

Furthermore, Equation (7) can also be written as the matrix form

$$M\ddot{q}+C\dot{q}+K_{e}q+K_h\mu \left({\dot{u}}_b\right)z_b=DQ\left(t\right)$$

(8)

in which

$$M=\left[\begin{array}{ccc}{\Lambda }_l+r_r{r}_r^{\mathrm{T}}& r_r& r_r{\Gamma }_u{\Lambda }_u\\ & 1& {\Gamma }_u{\Lambda }_u\\ \mathrm{sym}& & {\Lambda }_u\end{array}\right], D=\left[\begin{array}{ccc}{\Lambda }_l& r_r& {\mathbf{0}}_{r\times s}\\ 0& 1& 0\\ {\mathbf{0}}_{s\times r}& {\mathbf{0}}_{s\times 1}& {\Lambda }_u\end{array}\right];$$

$${\Lambda }_l=\mathrm{diag}\left(r_{l1},\dots ,r_{lr}\right), {\Lambda }_u=\mathrm{diag}\left(r_{u1},\dots ,r_{us}\right);$$

$${\Gamma }_u=\left[{\Gamma }_{u1}, \dots , {\Gamma }_{us}\right]; q={\left[q_l^{\mathrm{T}} u_b q_u^{\mathrm{T}}\right]}^{\mathrm{T}};$$

$$C=\mathrm{diag}\left(2{\omega }_{l1}{\zeta }_{l1}{r}_{l1},\dots , 2{\omega }_{lr}{\zeta }_{lr}{r}_{lr}, 2{\omega }_b{\zeta }_b, 2{\zeta }_{u1}{\omega }_{u1}{r}_{u1},\dots ,2{\zeta }_{us}{\omega }_{us}{r}_{us}\right);$$

$${\mathbf{K}}_{\mathit{e}}=\mathrm{diag}\left({\omega }_{l1}^2{r}_{l1},\dots , {\omega }_{lr}^2{r}_{lr}, {\omega }_b^2, {\omega }_{u1}^2{r}_{u1},\dots ,{\omega }_{us}^2{r}_{us}\right); K_h={\left[\begin{array}{ccc}{\mathbf{0}}_{1\times r}& g& {\mathbf{0}}_{1\times s}\end{array}\right]}^{\mathrm{T}};$$

$$Q\left(t\right)={\left[Q_{l1}\left(t\right), \dots , Q_{lr}\left(t\right), Q_b\left(t\right), Q_{u1}\left(t\right), \dots , Q_{us}\left(t\right)\right]}^{\mathrm{T}}$$

(9)

where ${\mathit{r}}_r$ is the r-dimension unit column vector.

The state–space form of the equation of motion is further given by

$$\dot{\nu }=g\left(\nu \right)+D_{0}Q\left(t\right)$$

(10a)

$$\nu =\left[\begin{array}{c}q\\ \dot{q}\\ z_b\end{array}\right]; g\left(\nu \right)=\left[\begin{array}{c}\dot{q}\\ -M^{-1}C\dot{q}-M^{-1}{K}_{e}q-M^{-1}{K}_h\mu \left({\dot{u}}_b\right)z_b\\ \left({\dot{u}}_b-\gamma \left|{\dot{u}}_b\right|{\left|z_b\right|}^{\eta -1}{z}_b-\beta {\dot{u}}_b{\left|z_b\right|}^{\eta }\right)/Y\end{array}\right]; D_0=\left[\begin{array}{c}\mathbf{0}\\ -M^{-1}D\\ \left[0\cdots 0\right]\end{array}\right]$$

(10b)

This nonlinear equation of motion can be solved by the Runge–Kutta method to obtain the structural response history.

The structural wind-induced response is commonly controlled by low modes. A few low-order dynamic parameters of the LS and US and the mechanical parameters of the isolation layer determine the structural system. Theoretically, through machine learning, it is possible to establish a mapping relationship between structural response and those parameters. Subsequently, the wind-induced response of buildings can also be predicted accurately [38].

2.2. Enhanced Simplified Model with Reduced DOFs

The equations of motion of inter-story isolated buildings in physical coordinates involve the structural dynamics model with full DOFs. The nonlinear time domain analysis of a structural system with a large number of DOFs could be inefficient. By modal truncation of the LS and US, the equations of motion in the modal coordinates can greatly reduce the DOFs of a system. The simplified model LrUs is defined as the combination of LS with first r modes, US with first s modes, and an isolation system (the simplest model is L1U1 with 3 DOFs). There are M × N simplified models of the original structure when the different r and s are combined. A reasonable simplified model is capable of accurately estimating the response of the original structure with the number of DOFs as few as possible. The author [29] proposed a method to select DOFs of the LS and US based on the relationship of generalized stiffness between the original structure and substructure:

$$K_{\mathrm{eff}, k}={{\displaystyle \sum }}_i^M{\overline{q}}_{li, k}^2{K}_{li}+{\overline{q}}_{b, k}^2{k}_{b,\mathrm{eff}}+{{\displaystyle \sum }}_j^N{\overline{q}}_{uj, k}^2{K}_{uj}(k=1,2,\dots ,M+N+1)$$

(11)

where $K_{\mathrm{eff}, k}$ is the k-th generalized stiffness of the original structure, $k_{b,\mathrm{eff}}$ is the equivalent stiffness of the isolation layer, and $K_{li}={\mathbf{\varphi }}_{ li}^{\mathrm{T}}{K}_l{\mathbf{\varphi }}_{ li}$ and $K_{uj}={\mathbf{\varphi }}_{ uj}^{\mathrm{T}}{K}_u{\mathbf{\varphi }}_{ uj}$ are the i-th generalized stiffness of LS and the j-th one of US, respectively. ${\overline{q}}_k={\left[{\overline{q}}_{l1, k} \dots {\overline{q}}_{li, k} \dots {\overline{q}}_{lM, k} {\overline{\mathrm{q}}}_{bk} {\overline{\mathrm{q}}}_{u1, k}\dots {\overline{q}}_{uj, k} \dots {\overline{q}}_{uN, k} \right]}^T$ is the mode vector corresponding to the undamped vibration of the original structure with full DOFs.

Using Equation (11), a reasonable reduced model is selected through the following procedure:

(1) The number of modes considered for the original structure is labelled as Np.

(2) For those Np modes, their important generalized stiffness contributions from the LS, US, and isolation layer need to be up to 98%. The highest mode of the LS and US that participates the cumulative operation is denoted as r_i and s_i (i = 1, 2,…, Np), respectively.

(3) Reasonable DOFs of the LS and US are r = max (r₁, r₂, …, r_Np) and s = max (s₁, s₂, …, s_Np), respectively.

For the nonlinear wind-induced vibration of a FPB isolated tall building, the isolation layer is always in a non-sliding phase or sliding phase. The equivalent stiffness of the isolation layer $k_{b,\mathrm{eff}}$ is taken as $W\left[\left(1/R\right)+\left({\mu }_{min}/Y\right)\right]$ or $W/R$ to the corresponding two phases. The final simplified model is the result with the consideration of the above two equivalent stiffnesses.

2.3. Statistical Linearization Method

In nonlinear random vibration analysis, the statistics of structural responses are computed by response history analysis. A large number of sample computations are required. Another frequency-domain-based statistical linearization method to obtain the statistics of structural responses directly is introduced as follows.

The nonlinear term $\mu \left({\dot{u}}_b\right)z_b$ in Equation (8) and the dimensionless hysteresis velocity term ${\dot{z}}_b$ in Equation (3c) are linearized as

$$\mu \left({\dot{u}}_b\right)z_b=b_1{\dot{u}}_b+b_2{z}_b$$

(12a)

$${\dot{z}}_b\approx b_3{u}_b+b_4{\dot{u}}_b+b_5{z}_b$$

(12b)

where coefficients $b_1$ and $b_2$ are derived according to the Gaussian distribution assumption [32]:

$$b_1=E\left[z_b\frac{d\mu \left({\dot{u}}_b\right)}{d{\dot{u}}_b}\right]=a\left({\mu }_{max}-{\mu }_{min}\right)\sqrt{\frac{2}{\pi }}{\rho }_{{\dot{u}}_b{z}_b}{\sigma }_{z_b}\Phi \left(\Lambda \right)$$

(13a)

$$b_2=E\left[\mu \left({\dot{u}}_b\right)\right]={\mu }_{max}-\left({\mu }_{max}-{\mu }_{min}\right)\exp \left(B^2\right)\mathrm{erfc}\left(\Lambda \right)$$

(13b)

where $\Phi \left(\Lambda \right)=1-\sqrt{\pi }\Lambda \exp ({\Lambda }^2)erfc\left(\Lambda \right)$, $\Lambda =\frac{a{\sigma }_{{\dot{u}}_b}}{\sqrt{2}}, \mathrm{erfc}\left(\Lambda \right)=1-\frac{2}{\sqrt{\pi }}{{\displaystyle \int }}_0^{\Lambda }\exp (-x^2)\mathrm{d}x$; and ${\sigma }_{{\dot{u}}_b}$, ${\sigma }_{z_b}$, and ${\rho }_{{\dot{u}}_b{z}_b}$ are the standard deviations (STDs) of ${\dot{u}}_b$ and $z_b$, as well as their correlation coefficient.

Baber and Wen [39] provided an analytic expression for $b_3$, $b_4$, and $b_5$:

$$b_3=E\left[\frac{\partial {\dot{z}}_b}{\partial u_b}\right]=0$$

(14a)

$$b_4=E\left[\frac{\partial {\dot{z}}_b}{\partial {\dot{u}}_b}\right]=A-\gamma F_1-\beta F_2$$

(14b)

$$b_5=E\left[\frac{\partial {\dot{z}}_b}{\partial {\dot{z}}_b}\right]=-\gamma F_3-\beta F_4$$

(14c)

in which

$$F_1=E\left[\frac{\partial \left|{\dot{u}}_b\right|}{\partial {\dot{u}}_b}{z}_b{\left|z_b\right|}^{n-1}\right]=\frac{{\sigma }_{z_b}^{\eta }}{\pi }\Gamma \left(\frac{\eta +2}{2}\right)2^{\eta /2}{I}_s$$

(15a)

$$F_2=E\left[{\left|z_b\right|}^n\right]=\frac{{\sigma }_{z_b}^{\eta }}{\sqrt{\pi }}\Gamma \left(\frac{\eta +1}{2}\right)2^{\eta /2}$$

(15b)

$$F_3=E\left[\frac{\partial z_b{\left|z_b\right|}^{n-1}}{\partial z_b}\left|{\dot{u}}_b\right|\right]=\frac{\eta {\sigma }_{{\dot{u}}_b}{\sigma }_{z_b}^{\eta -1}}{\pi }\Gamma \left(\frac{\eta +2}{2}\right)2^{\eta /2}[\left(2/\eta \right){\left(1-{\rho }_{{\dot{u}}_b{z}_b}^2\right)}^{\frac{\eta +1}{2}}+{\rho }_{{\dot{u}}_b{z}_b}{I}_s]$$

(15c)

$$F_4=E\left[\frac{\partial {\left|z_b\right|}^{\eta }}{\partial z_b}{\dot{u}}_b\right]=\frac{\eta {\rho }_{{\dot{u}}_b{z}_b}{\sigma }_{{\dot{u}}_b}{\sigma }_{z_b}^{\eta -1}}{\sqrt{\pi }}\Gamma \left(\frac{\eta +1}{2}\right)2^{\eta /2}$$

(15d)

where $I_s=2{{\displaystyle \int }}_l^{\pi /2}{\sin }^n\theta d\theta$, $l =\arctan \left(\sqrt{1-{\rho }_{{\dot{u}}_b{z}_b}^2}/{\rho }_{{\dot{u}}_b{z}_b}\right)$.

Accordingly, the corresponding linearized equation for an FPB inter-story isolated building is given by

$$\dot{v}=Rv+\overline{D}Q\left(t\right)$$

(16a)

$$v=\left[\begin{array}{c}q\\ \dot{q}\\ z_b\end{array}\right]; R=\left[\begin{array}{ccc}\mathbf{0}& I& {[0\cdots 0]}^T\\ -M^{-1}{K}_e& -M^{-1}{C}_{\mathrm{eff}}& -M^{-1}{K}_h{b}_2\\ \left[\begin{array}{ccc}{\mathbf{0}}_{1\times r}& b_3& {\mathbf{0}}_{1\times s}\end{array}\right]& \left[\begin{array}{ccc}{\mathbf{0}}_{1\times r}& b_4& {\mathbf{0}}_{1\times s}\end{array}\right]& b_5\end{array}\right]; \overline{D}=\left[\begin{array}{c}\mathbf{0}\\ M^{-1}D\\ \left[0\cdots 0\right]\end{array}\right]$$

(16b)

where $C_{\mathrm{eff}}=C+diag\left(\left[\begin{array}{ccc}{\mathbf{0}}_{1\times r}& b_{1}g& {\mathbf{0}}_{1\times s}\end{array}\right]\right)$.

The statistics of the steady-state response can be determined by

$$E\left[{\mathrm{vv}}^{\mathrm{T}}\right]={{\displaystyle \int }}_0^{+\infty }{S}_{vv}\left(\omega \right)d\omega$$

(17a)

$$S_{vv}\left(\omega \right)={(i\omega I-R)}^{-1}\overline{D}T{S}_{PP}\left(\omega \right)T^T{\overline{D}}^{\mathrm{T}}{(-i\omega I-R^{\mathrm{T}})}^{-1}$$

(17b)

$$T=\left[\begin{array}{ccc}{(M_{lr}^{*})}^{-1}{\Phi }_l^{\mathrm{T}}& {\mathbf{0}}_{r\times 1}& {\mathbf{0}}_{r\times N}\\ {\mathbf{0}}_{1\times M}& 1& r_N^{\mathrm{T}}\\ {\mathbf{0}}_{s\times M}& {\mathbf{0}}_{s\times 1}& {(M_{us}^{*})}^{-1}{\Phi }_u^{\mathrm{T}}\end{array}\right]$$

(17c)

$$M_{lr}^{*}=diag\left({\mathbf{\varphi }}_{ l1}^{\mathrm{T}}{M}_l{\mathbf{\varphi }}_{ l1} {\mathbf{\varphi }}_{ l2}^{\mathrm{T}}{M}_l{\mathbf{\varphi }}_{ l2},\dots {\mathbf{\varphi }}_{ lr}^{\mathrm{T}}{M}_l{\mathbf{\varphi }}_{ lr}\right)$$

(17d)

$$M_{us}^{*}=diag\left({\mathbf{\varphi }}_{ u1}^{\mathrm{T}}{M}_u{\mathbf{\varphi }}_{ u1} {\mathbf{\varphi }}_{ u2}^{\mathrm{T}}{M}_u{\mathbf{\varphi }}_{ u2},\dots {\mathbf{\varphi }}_{ us}^{\mathrm{T}}{M}_u{\mathbf{\varphi }}_{ us}\right)$$

(17e)

where $S_{PP}\left(\omega \right)$ is the PSD matrix of the story wind load vector $P={[P_l^{\mathrm{T}} P_b P_u^{\mathrm{T}}]}^{\mathrm{T}}$.

In the statistical linearization method, Equation (16) always contains the unknown parameters $b_1$, $b_2,b_4$, and $b_5$. These parameters depend on the STD of responses ${\dot{u}}_b$ and $z_b$, as well as their correlation coefficient. Therefore, an iterative computation is needed. These initial values of parameters can be taken as $b_1=0,b_2={\mu }_{min},b_4=1/Y$, and $b_5=0$.

3. Building Example

3.1. Building Description

For comparison, the fixed-base, base-isolated and inter-story isolated tall buildings, shown in Figure 2, are involved in this study. They have identical building dimensions, with a total height of $H=202 \mathrm{m}$ and square section of $B=D=0.2 H$. The story heights of those three building are 4 $\mathrm{m}$. The 2 m rigid foundation and isolation layer are set for fixed-base or isolated buildings. The first story of the fixed-base and base-isolated buildings is supported by rigid foundation and isolation bearings, respectively. Thus, all the buildings have 51 stories.

For the fixed-base building, the mass of each story is $m_i=1.2288\times {10}^6 \mathrm{kg}$ according to the building density ${\rho }_s=192 \mathrm{kg}/{\mathrm{m}}^3$. The first frequency is $f_{01}=0.20 \mathrm{Hz}$. The story stiffness is determined by use of the linear first mode shape, i.e., ${\varphi }_{01}\left(z\right)=z/H$. According to eigenvalue analysis, the first three order frequencies of the structure are 0.20, 0.49, and 0.77 $\mathrm{Hz}$, respectively. The building adopts Rayleigh damping with the first and third modal damping ratios ${\zeta }_{01}={\zeta }_{03}=1\%$ to determine the damping matrix. The eigenvalue analysis indicates that the first three modal damping ratios of a fixed-base building are 1.00%, 0.83%, and 1.00%, respectively.

A base-isolated building is regarded as a flexible fixed-base building which replaces the rigid foundation by the isolation layer. Ryan and Polanco [34] indicated that a stiffness-proportional damping matrix is appropriate for the US of base-isolated buildings. The first modal damping ratio ${\zeta }_{01}=1\%$ is calibrated to define this damping matrix. For the isolation system, the mass of the isolation slab is equal to that of one story of the US, i.e., $m_b=m_i$. The radius of the circular sliding surface of the FPB is $R=2 \mathrm{m}$. The parameters related to the friction coefficient are taken as ${\mu }_{min}=0.01$, ${\mu }_{min}/{\mu }_{min}=2$ and $a=23.6 \mathrm{s}/\mathrm{m}$. Additionally, it is assumed that the dissipative energy of the isolation system is only from the friction mechanism, and the linear damping ratio is ${\zeta }_b=0$.

An inter-story isolated building consists of the isolation layer, LS, and US. The parameters of the isolation system are the same as those in a base-isolated building. In fact, there is no uniform regulation for the preliminary structural design for the LS and US. Two design schemes to define the LS and US are introduced as follows:

Scheme A: Similar to the base-isolated building, the isolation layer is placed in any middle story of the fixed-base building, instead of the base. The story mass and stiffness of the LS and US are determined by the fixed-base building.

Scheme B: The LS and US are independent of each other and are designed separately. Specifically, the masses of the LS and US are calculated based on the building density, ${\rho }_l={\rho }_u=192 \mathrm{kg}/{\mathrm{m}}^3$. The first modal frequencies of the LS and US are $f_{l1}=$10/M and $f_{u1}=10/$N Hz, respectively. By use of the linear first mode shape, the story stiffness of each part is obtained.

The damping matrices of the LS and US are determined by their respective modal damping ratios in two design schemes. For the LS, the damping matrix $C_l$ is calculated according to ${\zeta }_{l1}={\zeta }_{l3}=1\%$. The damping matrix of the US $C_u$ is obtained through ${\zeta }_{u1}=1\%$.

In scheme A, a fixed-base tall building with a specified height is assumed, the to-be-built inter-story isolated tall building is designed by installing an isolation layer in the middle of this fixed-base building. Similar to base-isolated buildings, this structural design of an inter-story isolated tall building only needs to consider the dynamic characteristics of the fixed-base building and the mechanical parameters of the isolation layer. The design method is simple, and the integrity of the building is great. Scheme B is more flexible, especially when the characteristics of the LS and US are significantly different. It is suitable for isolated buildings on large bottom podiums or over tracks and adding-story-adding retrofit on existing buildings by isolation bearings.

3.2. Modal Characteristics

3.2.1. Scheme A

The dynamic model of an inter-story isolated tall building has M + 1 + N = 51 degrees of freedom, denoted as 51DOFs. According to Section 2.2, a method to select the simplified model of the original building with a 20-story LS (M = 20) is illustrated here. Figure 3 presents the contributions of generalized stiffness from the LS, isolation layer, and US to the first three generalized stiffnesses of the original building when the isolation layer is in the non-sliding and sliding phases. It can be seen that the generalized stiffness of the original building is contributed by only a few low modes of the LS and US. For example, the first modes of the LS and US contribute 91.24% to the first generalized stiffness of the original building, as the isolation layer in the non-sliding phase. This data is up to 92.86% in the sliding phase. If the first four DOFs of the LS and US are taken, the first three modes of the original building have an estimation with an accuracy up to 98% in both the non-sliding and sliding phases. Therefore, the reasonable simplified building model is L4U4.

The location of the isolation layer is labeled as M, i.e., the number of stories of the LS. The base-isolated building is represented by M = 0.

Table 1. Selected DOFs of the simplified models for scheme A.

Isolation Location M		0	5	10	15	20	25	30	35	40	45
Non-sliding	$r_i$	--	2	3	4	4	6	6	6	8	8
	$s_i$	3	4	4	4	4	2	3	3	2	1
Sliding	$r_p$	--	1	2	3	4	4	4	4	6	6
	$s_p$	4	4	4	4	3	3	3	3	2	1
LrUs	r = max ($r_i$, $r_p$)	0	2	3	4	4	6	6	6	8	8
	s = max ($s_i$, $s_p$)	4	4	4	4	4	3	3	3	2	1

gives selective simplified models of inter-story isolated buildings with different isolation locations.

Figure 4 and Figure 5 show the variation in the first three frequencies and damping ratios for the simplified model LrUs with the different isolation location when the isolation layer is in the non-sliding or sliding phase. The results from 51DOFs and L1U1 models are also provided for comparison. In addition, Figure 6 and Figure 7 depict the first three mode shapes of the inter-story isolated building with M = 20.

It is observed that scheme A does not alter the story stiffness of the LS and US with the elevation of the isolation layer. Therefore, the first modal frequency decreases slightly, while the second and third ones remain almost unchanged when the isolation layer is in the non-sliding phase. The modal frequencies in the sliding phase are always lower than those in the non-sliding phase. Their variation becomes more complex with the rise in the isolation layer.

With the elevation in the isolation layer, the first modal damping ratio initially decreases and then increases, while the second and third modal damping ratios reduce on the whole. In fact, the modal damping ratios of the original building (51DOFs) are the combination of those of the LS, isolation layer, and US [15].

In the non-sliding phase, the structural mode shapes are similar to those of the fixed-base building, while in the sliding phase, the linear stiffness ($W/R$) provided by the bearings is always smaller than the adjacent story stiffness of the LS and US, resulting in a clear drift at the isolation layer.

It is evident that the simplified model, LrUs, can accurately estimate the first three modal characteristics of the original building. Strictly speaking, even higher modes (such as the fourth or fifth mode) also have a great prediction. Because the wind-induced response of an isolated tall building is primarily controlled by the first three modes, the simplified model LrUs can entirely represent the model 51DOFs to compute wind-induced responses. The model L1U1 overestimates the first frequency and damping ratio and provides a different first mode shape. The estimations of the second and third modal characteristics are usually inaccurate.

3.2.2. Scheme B

When structural design adopts scheme B, the selection for the DOF of an appropriate simplified model is presented in

Table 2. Selected DOFs of the simplified models for scheme B.

Isolation Location M		0	5	10	15	20	25	30	35	40	45
Non-sliding	$r_i$	--	2	3	4	4	5	5	6	6	6
	$s_i$	3	4	4	4	3	3	2	1	1	1
Sliding	rp	--	2	3	3	4	4	5	5	5	6
	$s_p$	4	4	4	3	3	3	2	1	1	1
LrUs	r = max ($r_i$, $r_p$)	0	2	3	4	4	5	5	6	6	6
	s = max ($s_i$, $s_p$)	4	4	4	4	3	3	2	1	1	1

. Figure 8 and Figure 9 illustrate the first three modal frequencies and damping ratios. Because the first frequencies of the LS and US are inversely proportional to their respective numbers of stories, it is the softest for the whole structure when the isolation layer is positioned near the middle of the building. Therefore, the frequency decreases first and then increases with the isolation location M. The minimum first frequency is observed at around M = 25. Compared with the first frequency, the change in the second frequency with M has the opposite trend, and the third frequency has a more complex variation. The damping matrices for the LS and US are set independently. Therefore, the variations in modal damping ratios are similar to those in scheme A.

In scheme B, the story stiffness of the LS and US varies with the elevation of the isolation layer, and the LS story stiffness near the top is typically smaller than the US story stiffness adjacent to the bottom. Therefore, there is a significant mode displacement corresponding to stories close to the top of the LS (shows in Figure 10a). Nevertheless, the isolation layer stiffness in the sliding phase, $W/R$, is always smaller than its adjacent story stiffness of the LS and US, which leads to an obvious mode displacement at the isolation layer (shows in Figure 11).

Compared to scheme A, the model L1U1 causes a larger error for the prediction on the structural modal characteristics. Fortunately, the LrUs model still provides a satisfactory estimation.

4. Wind-Induced Response

This study focuses on the buffeting response of an inter-story isolated building to a fluctuating wind load. The wind load on the stories in Equation (5) is based on a quasi-static assumption. The dynamic wind–structure interaction is ignored, and the vortex-induced vibration is not considered here.

Because of the non-zero mean component, the along-wind load of the building is obviously different from the crosswind load. The mean wind load affects the displacement at the isolation layer before arriving to the steady state. However, this influence is significant only in the case of small sliding of the bearing under the low wind speed. When a distinct sliding occurs at the isolation layer, its displacement quickly reaches a steady state, and the along-wind response can be separately computed for the static response under the mean load and the dynamic response under the fluctuating load [30]. There are consistent solution procedures and similar structural response characteristics for along-wind and crosswind fluctuating responses. In the following discussion, the crosswind vibration of an inter-story isolated building will be focused on to analyze the wind-induced response characteristic and validate the effectiveness of the statistical linearization method.

The building locates in II terrain category with wind profile coefficient ${\alpha }_s=0.15$. A decay factor $k_y=5$ is used in crosswind PSDs of the story wind loads. The PSD of the crosswind base bending moment coefficient is determined from reference [35] with its STD ${\sigma }_{CM}=0.16$, as shown in Figure 12. Using the spectral representation method, 100 history samples with a duration of 15 min for the story wind loads are simulated. The PSD of the base moment coefficient obtained through stochastic simulation is consistent with that from the Japanese code. This study focuses on the statistics of structural responses in the steady state, and the first 5 min of the response history are discarded to eliminate the transient effect in the following analysis.

4.1. Dynamic Response of Buildings

4.1.1. Scheme A

For the isolation location M = 10, 20, 30, and 40, the STDs of responses, including displacement, drift, acceleration, and shear force, of inter-story isolated buildings designed through scheme A to crosswind load with $U_H=50 \mathrm{m}/\mathrm{s}$ are portrayed in Figure 13. The corresponding results for a fixed-base building and the base-isolated building (M = 0) are also plotted, which are comparable with those in reference [28]. The STDs of the top displacements of the fixed-base building in this study and in reference [28] are 0.26 and 0.23 m, respectively; and the STDs of the base shear forces of the base-isolated buildings shown in Figure 13d and in that reference are 7.84 × 10⁶ and 8.12 × 10⁶ kN, respectively. It is observed that the inter-story drift angle of the fixed-base building is 0.263/202 = 0.0013 < 1/550 (the limitation of the inter-story drift angle of the reinforced concrete frame structure [40]). The drift at the isolation layer increases with its elevation, while the story drifts of the LS and US remain almost at the same level. The story drifts of the LS adjacent to the isolation layer have a noticeable change, which leads to an obvious variation in the shear forces of corresponding stories. Apart from the isolation layer being positioned near the top, story accelerations change a little with the isolation location. The friction-induced energy dissipation of bearings increases the damping of the structural system [30,31]. Therefore, the responses of isolated buildings reduce significantly compared with those of the fixed-base building. For instance, the acceleration at the building top decreases by 44% and the shear force at the building base declines by 39% when the isolation location is M = 30.

By use of the LrUs and L1U1 models,

Table 3. Structural response STDs by the LrUs and L1U1 models (scheme A).

M		0	5	10	15	20	25	30	35	40	45
Du top (10−2 m)	LrUs	14.63	13.01	11.58	10.19	8.95	7.68	6.32	4.90	3.55	2.08
	L1U1	14.84	13.16	11.48	10.04	8.72	7.40	6.08	4.77	3.45	2.04
Di (10−2 m)	LrUs	1.55	1.73	1.93	2.13	2.38	2.67	3.04	3.49	4.16	5.41
	L1U1	1.53	1.67	1.80	1.94	2.11	2.35	2.68	3.13	3.82	5.21
Dl top (10−2 m)	LrUs	--	1.51	3.07	4.54	6.18	7.76	9.34	11.08	13.24	16.66
	L1U1	--	1.35	2.53	3.71	4.92	6.14	7.47	9.07	11.19	14.63
Au top (m·s−2)	LrUs	0.23	0.23	0.24	0.24	0.25	0.26	0.26	0.26	0.29	0.33
	L1U1	0.19	0.19	0.19	0.20	0.20	0.21	0.22	0.22	0.25	0.29
Al top (m·s−2)	LrUs	--	0.07	0.08	0.09	0.11	0.12	0.13	0.14	0.17	0.22
	L1U1	--	0.06	0.06	0.09	0.10	0.10	0.11	0.12	0.14	0.19
Vu base (106 N)	LrUs	7.84	7.47	7.10	6.59	6.13	5.43	4.71	3.90	2.94	1.75
	L1U1	7.79	6.71	6.03	5.42	4.83	4.22	3.56	2.88	2.17	1.39
Vl base (106 N)	LrUs	--	6.94	7.37	7.30	8.08	8.05	7.78	7.88	8.18	9.09
	L1U1	--	9.48	9.27	9.12	8.98	8.81	8.70	8.71	8.92	9.56

summarizes the STDs of some important responses of isolated buildings with different isolation locations. These responses include top displacement of the LS, drift at the isolation layer, top displacement of the US relative to isolated layer, top accelerations of the LS and US, as well as base shear forces of the LS and US, which are denoted as D_{l top}, D_i, D_{u top}, A_{l top}, A_{u top}, V_{l base}, and V_{u base}, respectively. The ratios of structural response STDs obtained from the L1U1 and LrUs models are shown in Figure 14. Figure 15 provides the PSD of D_{l top}, D_{u top}, A_{l top}, and A_{u top} normalized by their variances as M = 0, 10, 20, 30, and 40.

It can be found that the structural displacement is primarily dominated by the first mode, while the acceleration is controlled by the first three modes. As the isolation layer is raised, A_l top is gradually dominated by the first mode, while the first mode is always the main contribution to A_{u top}. Due to the inaccurate estimation on modal characteristics, the structural responses provided by L1U1 show errors more or less. Specifically, V_{l base} is overestimated and other responses are underestimated. D_{l top} is more prone to underestimation than D_{u top}.

4.1.2. Scheme B

With scheme B, the STDs of displacement, drift, acceleration, and shear force along the stories of the inter-story isolated building are depicted in Figure 16.

Table 4. Structural response STD according to the LrUs and L1U1 models (scheme B).

M		0	5	10	15	20	25	30	35	40	45
Du top (10−2 m)	LrUs	14.63	12.16	10.94	9.74	8.48	6.76	4.68	2.72	1.18	0.33
	L1U1	14.84	12.20	10.29	8.56	6.80	5.08	3.50	2.12	1.02	0.28
Di (10−2 m)	LrUs	1.55	1.78	2.36	3.11	4.11	5.06	5.90	6.22	6.27	7.01
	L1U1	1.53	1.68	1.99	2.35	2.75	3.20	3.70	4.32	5.15	6.25
Dl top (10−2 m)	LrUs	--	2.12	6.03	11.25	17.73	23.99	28.16	29.44	27.68	25.76
	L1U1	--	1.78	4.19	6.79	9.42	11.97	14.33	16.34	17.96	19.26
Au top (m·s−2)	LrUs	0.23	0.23	0.25	0.27	0.30	0.33	0.35	0.35	0.34	0.37
	L1U1	0.19	0.18	0.19	0.20	0.21	0.22	0.23	0.25	0.27	0.30
Al top (m·s−2)	LrUs	--	0.10	0.12	0.15	0.19	0.23	0.26	0.28	0.28	0.31
	L1U1	--	0.07	0.08	0.11	0.13	0.14	0.16	0.18	0.20	0.23
Vu base (106 N)	LrUs	7.84	7.59	8.09	8.56	8.94	8.77	7.69	5.93	3.88	1.94
	L1U1	7.79	6.72	6.40	6.10	5.68	5.12	4.45	3.66	2.73	1.62
Vl base (106 N)	LrUs	--	6.95	9.75	9.29	10.96	11.59	12.31	11.04	10.23	9.76
	L1U1	--	10.39	11.18	11.71	12.00	12.08	11.97	11.65	11.16	10.61

summarizes the STDs of important responses computed by simplified models L1U1 and LrUs. The ratios of response STDs by those two models are illustrated in Figure 17. The normalized PSDs of D_{l top}, D_{u top}, A_{l top}, and A_{u top} are displayed in Figure 18.

It is observed that the stories near the top of the LS exhibit a significant drift with the relatively low story stiffness, while the drift of the US can be effectively mitigated. With the rise in isolation location, the drift at the building top decreases greatly, and the largest drift at the isolation layer continues to grow. Because the structural response is dominated by the first mode, and the first modal frequency and damping ratio of the inter-story isolated building are always lower than those of the base-isolated building (as shown in Figure 8a and Figure 9a), the displacement, acceleration, and shear forces of the former are consistently larger than those of the latter. The maximal wind-induced response appears at M = 30, at which the acceleration and shear force approach the results of the fixed-base building. Similarly, due to the additional damping from the isolation layer, the story acceleration and shear force of the isolated building are, on the whole, less than those of the fixed-base building.

In scheme B, the estimation of structural dynamic properties by L1U1 is worse than that in scheme A, resulting in less accurate predictions of structural responses. For example, D_{l top} is underestimated by 50% when M = 30. With the comparison between Figure 15 and Figure 18, scheme B provides a more flexible building structure, and its structural response is predominantly contributed by the first and second modes.

It should be pointed that for a known site and specified height of the inter-story isolated tall building, the location of the isolation layer directly affects the dynamic characteristics of the LS and US, thereby affecting the dynamic response of the building. The optimal location of the isolation layer can be further determined through genetic algorithm-based single or multi-objective optimization methods [21,22]. The optimization objectives (related to the building’s intended use and safety) such as minimizing the displacement of the isolation layer and the acceleration of the LS and US will directly affect the location of the isolation layer.

4.2. Accuracy of the Statistical Linearization Method

Taking the inter-story isolated building (M = 20) designed by scheme A as an example, Figure 19 compares the response STDs computed by response history analysis (RHA) and statistical linearization method (SLM) in the wind speed range $U_H=20~60 \mathrm{m}/\mathrm{s}$. It can be observed that the SLM slightly overestimates displacement at the isolation layer at a high wind speed, while it underestimates the accelerations at the top of the LS, somewhat. The other structural responses have a great prediction.

Through the eigenvalue analysis of state-space equation, i.e., Equation (16), the equivalent dynamic properties of the inter-story isolated building with nonlinearity can be obtained. Figure 20 provides the variation in the first three frequencies and damping ratios of the equivalent linear system with the wind speed. It is clear that the first and third frequencies of the building decrease slightly with the growth of $U_H$, while the second frequency is almost unchanged. This variation is consistent with the result in Figure 4a where frequencies change with the bearing state from the non-sliding phase to the sliding phase. The friction-induced energy dissipation brings about a significant increase in the first and third modal damping ratios. It is the additional damping at the isolation layer, the wind-induced responses of inter-story isolated tall buildings are reduced, on the whole.

It should be noted that the SLM provides an equivalent linear system to represent a nonlinear system by minimizing the mean square error between these two systems. The accuracy of the SLM depends on the degree of nonlinearity of the responses, i.e., the effectiveness of the SLM is related to the structural nonlinearity and the input load.

5. Conclusions

This study conducted the wind-induced vibration analysis of an inter-story isolated tall building with FPBs. The nonlinear time domain analysis framework and statistical linearization method to obtain the wind-induced response of the building are built by a simplified model with reduced DOFs. Considering the independence of the LS and US, two structural design schemes for FPB inter-story isolated tall buildings are provided. Their dynamic properties are analyzed, and the crosswind responses are compared. Finally, the accuracy of the statistical linearization method is confirmed. Some important conclusions are listed as follows:

1. The main factors affecting structural responses include low-order modal parameters of the LS and US (mass ratios, frequency ratios, damping ratios), and the friction coefficient and sliding plate radius of the FPB, as well as wind loads.

2. Although the wind-induced vibration of inter-story isolated tall buildings is controlled by low modes, the simplified model L1U1 consisting of the first mode of the LS and US cannot accurately model the first three dynamic characteristics of the original building, resulting in a significant over- and under-estimation of the structural responses. For instance, the top displacement of the LS is underestimated by nearly 50% in the case of structural design scheme B. The proposed simplified model LrUs can predict the first three modes precisely, and the structural response also has an accurate estimation.

3. The story stiffness of the LS and US in design scheme A is determined by the corresponding fixed-base building. The existence of the isolation layer affects the response of stories near the isolation layer. The displacement at isolation layer increases with its elevation, other responses are not significantly affected by the isolation location. The top acceleration and base shear force of the inter-story isolated building with M = 30 reduce by 44% and 30%, respectively, which attributes to the additional damping from friction-induced energy dissipation.

4. The story stiffness of the LS and US in design scheme B is defined independently. Its dynamic characteristic and wind-induced response vary greatly with the isolation location. When the isolation layer is near to middle position (M = 30), the smallest first modal frequency leads to the highest structural response and the worst isolation performance. Overall, scheme B makes the building structure more flexible than scheme A, and gives the greater wind-induced vibration.

5. Except for a slight underestimation of the top acceleration of the LS, and an overestimation of the displacement of the isolation layer at the high wind speed, the statistical linearization is accurate to predict other structural responses.

This study investigates the wind-induced response of FPB inter-story isolated buildings from a theoretical perspective. In future research, it is necessary to conduct a corresponding experimental study to confirm the accuracy and reliability of the proposed methods and design schemes. In addition, the response of a hybrid isolation system consisting of rubber bearings and friction pendulum bearings to three-dimensional synchronous wind loads also deserves investigation.