Equivalent SDOF Models for Estimating Isolation-Layer Energy Dissipation in Base-Isolated Tall Buildings under Strong Winds

Abstract

An isolation layer composed of isolators and steel dampers in base-isolated tall buildings can dissipate wind-induced energy via repeated elasto-plastic deformation. Its energy dissipation can be used to estimate its wind-induced responses and the fatigue damage in the steel dampers. Computationally costly time history analyses using multi-degree-of-freedom (MDOF) models suggest that some structural parameters influence the isolation-layer energy dissipation. However, using common single-degree-of-freedom (CS) models cannot fully capture such influences (e.g., those caused by the damping ratio and the natural period of the upper structure). Hence, this paper proposes a more accurate new equivalent single-degree-of-freedom (ES) model to estimate the isolation-layer energy dissipation in base-isolated tall buildings under strong winds. The ES model considers the influence of structural parameters and uses the first mode shapes of the MDOF models. It is as computationally efficient as, but is more accurate than, the CS model. The results indicate that it can estimate the isolation-layer energy dissipation as closely as MDOF models of base-isolated tall-building under strong winds.

1. Introduction

Recently, base isolation systems have been applied to many tall buildings to ensure their structural safety during earthquakes [1,2,3]. They are mainly composed of (i) isolators such as rubber bearings and (ii) dampers such as viscous, viscoelastic, and steel dampers [4]. Steel dampers are most commonly used in base isolation systems due to their low cost and high performance [5,6]. For example, they require little maintenance and can dissipate seismic energy via repeated elasto-plastic deformation to minimize structural damage. In the seismic design of base-isolated tall buildings, structural designers can decrease the yield shear force coefficient of the steel dampers to promote the building’s seismic performance because these dampers yield early, so they can dissipate a great amount of seismic energy after yielding.

In earthquake- and typhoon-prone areas such as Japan, the wind-resistant performance of base-isolated tall buildings should also be addressed [7,8,9,10]. However, for those base-isolated tall buildings equipped with steel dampers, promoting their seismic performance by, for example, decreasing the yield shear force coefficient of the steel dampers means demoting their wind-resistant performance. This is because these dampers may result in significant elasto-plastic responses in the base isolation layer, as they yield easily during strong winds. For example, the maximum deformation of the base isolation layer can reach around 10 cm in typhoon simulations [11]. Moreover, a study has shown that isolators with nonlinear damping perform better at a high frequency than with linear damping [12]. Another more important aspect is that wind forces have a much longer duration than seismic forces, which may lead to the accumulation of fatigue damage in yielding dampers. For example, the typhoon simulation results show that the fatigue damage in steel dampers can even be close to 1.0 in certain typhoon samples [13]. Suppose those fatigue-damaged dampers are not replaced in time: fatigue fracture may occur in those dampers during the next large earthquake [14,15], eventually leading to a decrease in the seismic performance of the building. However, only a few studies have been conducted on the wind-induced responses of base isolation layers, as well as the fatigue damage in steel dampers, because the current wind-resistant design in Japan is based on buildings behaving elastically under strong winds [16].

An energy balance method has been established by balancing the seismic energy input and isolation layer energy dissipation, suggesting that the seismic responses of the base isolation layer can be estimated as long as the isolation layer energy dissipation is obtained [17]. Similarly, the wind-induced responses of the base isolation layer can also be estimated if the isolation layer energy dissipation is obtained [18,19,20,21,22,23]. On the other hand, the isolation layer energy dissipation will be very close to the steel damper energy dissipation when the duration of the wind forces is long enough [23]. In relevant fields, D. Rigon et al. estimated the multiaxial fatigue behavior of C45 steel specimens by using the energy dissipation [24]. From the above study, we can infer that damper fatigue damage is highly relevant to the isolation layer energy dissipation. Hence, studying the isolation layer energy dissipation can help estimate the wind-induced responses of the base isolation layer and fatigue damage in the steel dampers.

Multi-degree-of-freedom (MDOF) models with multi-story wind forces are commonly analyzed in time history analyses to effectively estimate the isolation layer energy dissipation in base-isolated tall buildings under strong winds. However, time history analysis is time-consuming because MDOF models have many parameters [11,13], and multi-story wind forces should consider ensemble methods [25,26]. Hence, for efficient time history analysis, proposing a single-degree-of-freedom (SDOF) model with only a one-story force that can also be used to estimate the isolation layer energy dissipation is required. Such efficiency is highly desired, especially when assessing the resiliency of a structure after a major event [27,28]. In previous studies, two types of SDOF models have been proposed in earthquake engineering. One is a common single-degree-of-freedom (CS) model assuming the upper structure is a rigid body (Appendix A). The CS model does not consider the upper structure’s characteristics (natural period and damping ratio), and all its structural parameters are calculated based on the assumption of a rigid upper structure. The other is a single-degree-of-freedom model proposed by Kasai et al. [29]. It considers the upper structure characteristics, but the calculation of its partial structural parameters (mass and wind force) is also based on the assumption of a rigid upper structure rather than a soft upper structure, the same as in the CS model. Obviously, the above two SDOF models are not appropriate for base-isolated tall buildings in wind engineering, and there is no appropriate SDOF model at present.

In view of the above, this paper aims to propose a new equivalent single-degree-of-freedom (ES) model to estimate the isolation layer energy dissipation in base-isolated tall buildings under strong winds. The ES model must meet both of the following requirements: (i) the upper structure characteristics are considered and (ii) the calculation of all the structural parameters is based on a soft upper structure. Structural engineers need this ES model to decrease the computing time for efficient wind-resistant design like the CS model and to accurately estimate the isolation-layer energy dissipation like the MDOF model. This paper first investigated the influences of the structural parameters on the isolation-layer energy dissipation in MDOF models using time history analyses (Section 2). Next, considering these influences, a modeling method for the ES model based on the first mode of the MDOF model was described (Section 3). Then, the accuracy of the isolation-layer energy dissipation in the ES model was verified by comparing the isolation-layer energy dissipation in the ES model and MDOF model (Section 4). Finally, conclusions and related remarks are given (Section 5).

2. Isolation-Layer Energy Dissipation in MDOF Models

The base-isolated tall building considered in this paper (Figure 1a) was idealized into a theoretical shear-type MDOF model (Figure 1b). Its upper structure, with assumed dimensions of height $H_u=100 \mathrm{m}$ ($u$ denotes upper structure), width $B_u=25 \mathrm{m}$, and depth $D_u=25 \mathrm{m}$, is expressed using an elastic 10-degree-of-freedom model (herein viewed as a 10-story model). Its isolation layer, composed of rubber bearings and steel dampers, is expressed using an elasto-plastic SDOF model. More details about the MDOF model are provided in Section 2.1, Section 2.2, Section 2.3 and Section 2.4.

2.1. Wind Forces of MDOF Models

The 500-year return period wind speed at the top of the building $U_{500y}=50.41 \mathrm{m}/\mathrm{s}$ in Figure 1a was obtained based on a basic wind speed of 36 m/s and surface roughness category III [30]. The wind force at the $i$-th story of the upper structure $F_{ui}$ ($i=1, 2,\dots , 10$) in Figure 1b was calculated for a 700-s duration at a 0.05-s time interval, using $U_{500y}$ and the story wind force coefficient obtained from the wind tunnel test [30]. Note that the wind force at the isolation layer $F_b=0$ ($b$ denotes the base isolation layer).

A total of 40 cases of wind forces (herein referred to as Force 01~40) in the along-wind and across-wind directions are considered in this paper [25,26]. As an example, Figure 2 shows the wind forces at the 10th story of the upper structure $F_{u10}$ (Force 01), in which the along-wind direction has a mean component while the across-wind direction has none. Note that the first and last 50-s wind forces are envelopes used to avoid transient responses of the MDOF models in the time history analysis. Therefore, they are not considered in the calculation of the isolation-layer energy dissipation, as will be presented in Section 2.4.

2.2. Modeling Method for the MDOF Models

For the MDOF model in Figure 1b, the mass of the $i$-th story of the upper structure $m_{ui}$ is expressed as:

$$m_{ui}=\frac{{\rho }_u·H_u·B_u·D_u}{10}\left(i=1,2,\dots ,10\right)$$

(1)

where ${\rho }_u$ is the density of the upper structure of the base-isolated tall building (=250 kg/m³).

Using $m_{ui}$ as obtained from Equation (1), the stiffness of the $i$-th story of the upper structure $k_{ui}$ can be expressed by [31]:

$$k_{ui}=\left\{\begin{array}{l}\frac{{\left(2\pi /T_u\right)}^2·m_{u1}·{\phi }_{u1}+k_{u2}\left({\phi }_{u2}-{\phi }_{u1}\right)}{{\phi }_{u1}} \left(i=1\right)\\ \frac{{\left(2\pi /T_u\right)}^2·m_{ui}·{\phi }_{ui}+k_{u\left(i+1\right)}\left({\phi }_{u\left(i+1\right)}-{\phi }_{ui}\right)}{{\phi }_{ui}-{\phi }_{u\left(i-1\right)}} \left(i=2,3,\dots ,9\right)\\ \frac{{\left(2\pi /T_u\right)}^2·m_{u10}·{\phi }_{u10}}{{\phi }_{u10}-{\phi }_{u9}} \left(i=10\right)\end{array}\right.$$

(2)

where $T_u$ is the natural period of the upper structure of the MDOF model (the values are presented in Section 2.3) and ${\phi }_{ui}$ is the mode shape of the $i$-th story in $\left\{{\phi }_u\right\}$. Here, $\left\{{\phi }_u\right\}$ is the first mode shape of the upper structure of the MDOF model, assumed as a straight distribution, which is commonly used in modeling.

Assuming the upper structure is a stiffness-proportional damping model, the damping coefficient of the $i$-th story of the upper structure $c_{ui}$ using $k_{ui}$ obtained from Equation (2) is given by:

$$c_{ui}=\frac{{\xi }_u·T_u·k_{ui}}{\pi }$$

(3)

where ${\xi }_u$ is the damping ratio of the upper structure of the MDOF model (values are presented in Section 2.3).

On the other hand, the mass of the isolation layer $m_b$ can be written as:

$$m_b={\rho }_b·B_u·D_u$$

(4)

where ${\rho }_b$ is the areal density of the isolation layer of the base-isolated tall building (=3644 kg/m²).

Furthermore, the other parameters of the isolation layer can be computed using the shear force–deformation relationship of the rubber bearing ($Q_f$-${\delta }_f$ relationship), steel damper ($Q_s$-${\delta }_s$ relationship), and isolation layer ($Q_b$-${\delta }_b$ relationship), as shown in Figure 3a–c. Here, $f$ and $s$ denote the rubber bearing and steel damper, respectively.

First, for the rubber bearing modeled as linearly elastic (Figure 3a), the stiffness $k_f$ is defined as [4]:

$$k_f=\frac{4{\pi }^2\left(m_b+\sum _{i=1}^{10}{m}_{ui}\right)}{T_f^2}$$

(5)

where $T_f$ is the isolated period of the MDOF model or the natural period of the MDOF model with a rigid upper structure and no steel damper (values are presented in Section 2.3).

Next, for the steel damper modeled using an elastic–perfectly plastic model (Figure 3b), the yield shear force $Q_{sy}$ ($y$ denotes the yield) and initial stiffness $k_{se}$ ($e$ denotes the elastic range) are described as follows [4]:

$$Q_{sy}=\left(m_b+\sum _{i=1}^{10}{m}_{ui}\right)g·{\alpha }_{sy}$$

(6)

$$k_{se}=\frac{Q_{sy}}{{\delta }_{sy}}$$

(7)

where $g$ is the gravitational acceleration, ${\alpha }_{sy}$ is the yield shear force coefficient of the steel damper in the MDOF model (values are presented in Section 2.3), and ${\delta }_{sy}$ is the yield deformation of the steel damper in the MDOF model (values are presented in Section 2.3).

Finally, for the isolation layer with bilinear behavior (Figure 3c) due to combining the rubber bearing (Figure 3a) and steel damper (Figure 3b), the initial stiffness $k_{be}$, post-yield stiffness $k_{bp}$ ($p$ denotes plastic range), yield deformation ${\delta }_{by}$, and yield strength $Q_{by}$ can be formulated, respectively, as follows [4]:

$$k_{be}=k_f+k_{se}$$

(8)

$$k_{bp}=k_f$$

(9)

$${\delta }_{by}={\delta }_{sy}$$

(10)

$$Q_{by}={\delta }_{by}·k_{be}$$

(11)

Note that for the isolation layer, the damping coefficient $c_b=0$ because the damping ratio is assumed to be ${\xi }_b=0$.

2.3. Structural Parameters and Their Influences on MDOF Models

According to the equations in Section 2.2, certain structural parameters determine the behavior of the MDOF model and therefore may subsequently influence the isolation-layer energy dissipation. These structural parameters and their values [32] are as follows:

(a)

Damping ratio of the upper structure

$${\xi }_u=1\%, 2\%, \mathrm{and} 5\%.$$

(b)

Natural period of the upper structure

$$T_u=1.5 \mathrm{s}, 2.0 \mathrm{s}, 2.5 \mathrm{s}, 3.0 \mathrm{s}, \mathrm{and} 3.5 \mathrm{s}.$$

(c)

Isolated period

$$T_f=4.0 \mathrm{s}, 4.5 \mathrm{s}, 5.0 \mathrm{s}, 5.5 \mathrm{s}, \mathrm{and} 6.0 \mathrm{s}.$$

(d)

Yield shear force coefficient of the steel damper

$${\alpha }_{sy}= 0.020, 0.025, 0.030, 0.035, \mathrm{and} 0.040.$$

(e)

Yield deformation of the steel damper

$${\delta }_{sy}= 1.5 \mathrm{cm}, 2.0 \mathrm{cm}, 2.5 \mathrm{cm}, 3.0 \mathrm{cm}, \mathrm{and} 3.5 \mathrm{cm}$$

The influences of these structural parameters on the MDOF model obtained from Section 2.2 are investigated in this section, and the findings are used for detailed discussions on the isolation-layer energy dissipation in Section 2.4.

Figure 4a shows the mass distribution for the upper structure obtained from Equation (1). The masses of all stories have the same value, and they are not influenced by the above five structural parameters. Figure 4b shows the stiffness distribution for the upper structure obtained from Equation (2). The stiffness decreases with the story level for each value of $T_u$, and the upper structure with a long $T_u$ shows a low stiffness. Among the five structural parameters, only $T_u$ has an influence on the stiffness distribution. Figure 4c shows the damping coefficient distribution for the upper structure obtained from Equation (3). For $T_u=3.0 \mathrm{s}$, the damping coefficient for the upper structure increases with an increasing ${\xi }_u$. The same can be said for the other values of $T_u$. Moreover, by comparing the damping coefficient for $T_u=3.0 \mathrm{s} \mathrm{and} 3.5 \mathrm{s}$, the upper structure with a longer $T_u$ shows a lower damping coefficient. Overall, only ${\xi }_u$ and $T_u$ can influence the damping coefficient for the upper structure.

Figure 5a–d show the influences of the different structural parameters ($T_u$, $T_f$, ${\alpha }_{sy}$, and ${\delta }_{sy}$) on the shear force–deformation relationship of the isolation layer ($Q_b$-${\delta }_b$ relationship) obtained from Equations (8)–(11). As indicated in Figure 5a, $T_u$ does not influence the $Q_b$-${\delta }_b$ relationship since it is not used for calculating any parameter of the isolation layer. The same can be said for the influence of ${\xi }_u$. As indicated in Figure 5b, $T_f$ has a small influence on both the elastic and plastic ranges. As $T_f$ increases, the initial stiffness decreases, leading to a decreasing yield shear force, and the post-yield stiffness also decreases. This is because the increase in $T_f$ results in a decrease in $k_f$ (Equation (5)), which then results in a decreasing $k_{be}$ (Equation (8)) and $k_{bp}$ (Equation (9)). As indicated in Figure 5c, ${\alpha }_{sy}$ has a significant influence on the elastic range but none on the plastic range. As ${\alpha }_{sy}$ increases, the initial stiffness increases, leading to an increasing yield shear force, while the post-yield stiffness remains unchanged. This is due to the fact that an increasing $k_{se}$ with an increasing ${\alpha }_{sy}$ (Equations (6) and (7)) can lead to an increase in $k_{be}$ (Equation (8)). As indicated in Figure 5d, ${\delta }_{sy}$ also has a significant influence on the elastic range but none on the plastic range. As ${\delta }_{sy}$ increases, the initial stiffness decreases, leading to a large yield shear force and yield deformation. The reason for this may be related to Equations (7) and (8).

2.4. Influences of the Structural Parameters on the Isolation-Layer Energy Dissipation

After performing time history analysis for the MDOF model (Section 2.2) subjected to the wind forces (Section 2.1) considering different values of the structural parameters (Section 2.3), the isolation-layer energy dissipation in the isolation layer of the MDOF model (per second [23]) ${\dot{W}}_b$ is obtained as follows:

$${\dot{W}}_b=\frac{1}{t_1-t_0}{\int }_{t_0}^{t_1}{Q}_b\cdot {\dot{\delta }}_{b}dt$$

(12)

where ${\dot{\delta }}_b$ is the derivative of ${\delta }_b$ with respect to time. As explained in Section 2.1, $t_0=50 \mathrm{s}$, and $t_1=650 \mathrm{s}$.

Figure 6a–d show the isolation-layer energy dissipation in the isolation layer of the MDOF model (per second) ${\dot{W}}_b$ obtained from Equation (12) considering the different values of the structural parameters. Note that each value of ${\dot{W}}_b$ is the ensemble average for Force 01~40. In the following, the ${\dot{W}}_b$ value in the across- and along-wind directions are discussed separately due to their different results.

In the across-wind direction, ${\dot{W}}_b$ increases significantly with the increase in $T_u$, as shown in Figure 6a. For each value of ${\xi }_u$, ${\dot{W}}_b$ increases by more than 6 times as $T_u$ ranges from 1.5 s to 3.5 s. Although $T_u$ has no influence on the $Q_b$-${\delta }_b$ relationship (Figure 5a), it influences the stiffness distribution for the upper structure (Figure 4b). It follows that the MDOF model with a soft upper structure (a long $T_u$) tends to have significant wind responses, including a large deformation of the base-isolation layer that results in a great amount of ${\dot{W}}_b$. In Figure 6b, ${\dot{W}}_b$ in the across-wind direction increased slightly with the increase in $T_f$. For $T_f=4~6 \mathrm{s}$, ${\dot{W}}_s$ ranges from 670.2 kN·cm/s to 871.0 kN·cm/s for ${\xi }_u=1\%$, from 434.5 kN·cm/s to 647.0 kN·cm/s for ${\xi }_u=2\%$, and from 136.4 kN·cm/s to 298.6 kN·cm/s for ${\xi }_u=5\%$. This slight trend in ${\dot{W}}_b$ is attributed to the slight decrease in the initial and post-yield stiffness of the isolation layer with an increasing $T_f$ (Figure 5b). In Figure 6c, ${\dot{W}}_b$ in the across-wind direction decreases significantly with the increase in ${\alpha }_{sy}$. ${\dot{W}}_b$ decreases by 71.8% for ${\xi }_u=1\%$, by 85.1% for ${\xi }_u=2\%$, and by 96.6% for ${\xi }_u=5\%$, as ${\alpha }_{sy}$ increases from 0.02 to 0.04. This significant decrease is due to the significant increase in the yield shear force of the isolation layer with an increasing $T_f$ (Figure 5c), leading to difficulty in dissipating the wind-induced energy with the damper yielding. In Figure 6d, ${\dot{W}}_b$ in the across-wind direction increases slightly with the increase in ${\delta }_{sy}$, which is similar to the trend observed in Figure 6b. The reason for this slight trend is that the isolation layer becomes soft in the elastic range by increasing ${\delta }_{sy}$ (Figure 5d) so that the isolation layer can dissipate more wind-induced energy. Moreover, ${\dot{W}}_b$ in the across-wind direction decreases significantly with the increase in ${\xi }_u$ as the other structural parameters ($T_u$, $T_f$, ${\alpha }_{sy}$, and ${\delta }_{sy}$) remain unchanged, as shown in Figure 6a–d. This is because the upper structure with a larger ${\xi }_u$ can dissipate more wind-induced energy, leading to less wind-induced energy that can be dissipated by the isolation layer. Summing up, in the across-wind direction, ${\xi }_u$, $T_u$, and ${\alpha }_{sy}$ show significant impacts on ${\dot{W}}_b$, which implies that a base-isolated tall building with a large ${\xi }_u$, a large $T_u$, and a small ${\alpha }_{sy}$ can easily produce a significant amount of ${\dot{W}}_b$. Therefore, structural designers should pay special attention to this aspect when conducting the wind-resistant design of base-isolated tall buildings. Furthermore, $T_f$ and ${\delta }_{sy}$ show minor impacts on ${\dot{W}}_b$ in the across-wind direction.

On the other hand, ${\dot{W}}_b$ in the along-wind direction is significantly small and varies negligibly with increasing structural parameters (${\xi }_u$, $T_u$, $T_f$, ${\alpha }_{sy}$, and ${\delta }_{sy}$), as shown in Figure 6a–d. The reason for this can be explained by the $Q_b$-${\delta }_b$ relationship for $t=50~650 \mathrm{s}$ shown in Figure 7a,b. For a specific MDOF model (${\xi }_u=2\%$, $T_u=2.5 \mathrm{s}$, $T_f=5.0 \mathrm{s}$, ${\alpha }_{sy}=0.03$, ${\delta }_{sy}=2.5 \mathrm{cm}$), the $Q_b$-${\delta }_b$ relationship in the along-wind direction (Figure 7b) shows much fewer and smaller hysteresis loops than those in the across-wind direction (Figure 7a). This is because the wind forces in the along-wind direction have mean components (Figure 2) in the $+{\delta }_b$ direction, which leads to the fact that there is no large wind force in the $-{\delta }_b$ direction and can make the isolation layer deform plastically in the $-{\delta }_b$ direction. To sum up, the above five structural parameters show negligible impacts on ${\dot{W}}_b$ in the along-wind direction.

3. Proposed Equivalent Single-Degree-of-Freedom Models

3.1. Modeling Concept

As found in Section 2.3 and Section 2.4, ${\xi }_u$ and $T_u$ influence the MDOF model and therefore subsequently influence the isolation-layer energy dissipation. This contradicts the common single-degree-of-freedom (CS) model, which cannot be influenced by ${\xi }_u$ and $T_u$ (Section 1 and

Table A1. Influences of structural parameters on the CS model.

	${\mathit{m}}^{\mathit{c}\mathit{s}}$	${\mathit{k}}_{\mathit{e}}^{\mathit{c}\mathit{s}}$	${\mathit{k}}_{\mathit{p}}^{\mathit{c}\mathit{s}}$	${\mathit{Q}}_{\mathit{y}}^{\mathit{c}\mathit{s}}$	${\mathit{\delta }}_{\mathit{y}}^{\mathit{c}\mathit{s}}$	${\mathit{\xi }}^{\mathit{c}\mathit{s}}$	${\mathit{F}}^{\mathit{c}\mathit{s}}$
${\xi }_u$	×	×	×	×	×	×	×
$T_u$	×	×	×	×	×	×	×
$T_f$	×	○	○	○	×	×	×
${\alpha }_{sy}$	×	○	×	○	×	×	×
${\delta }_{sy}$	×	○	×	○	○	×	×

○ denotes having influence and × denotes having no influence.

in Appendix A). In order to evaluate the isolation-layer energy dissipation as accurately as the MDOF model and with the computational efficiency of the CS model (Appendix A), this paper proposes a new equivalent single-degree-of-freedom (ES) model which considers ${\xi }_u$ and $T_u$ as well as $T_f$, ${\alpha }_{sy}$, and ${\delta }_{sy}$.

On the other hand, for an elastic MDOF model of a tall building, its elastic first mode model can show nearly equal wind-induced responses. In addition, several structural parameters of the elastic MDOF model, including the natural period and damping ratio, can affect its elastic first mode model. Based on these findings, for the elasto-plastic MDOF model considered in this paper, its elasto-plastic first mode model (herein referred to as the ES model) is likely to show a high accuracy for the isolation-layer energy dissipation corresponding to the structural parameters of the elasto-plastic MDOF model under strong winds. For this reason, the modeling method for the ES model proposed in this paper is based on the first mode of the elasto-plastic MDOF model, as will be discussed Section 3.3. Since the MDOF model considered in this paper has elasto-plastic behavior, its first mode shape can vary with the current stiffness (initial or post-yield stiffness) of the isolation layer. As such, the possible variation in the first mode shape is investigated first in Section 3.2.

3.2. First Mode Shapes of MDOF Models

When the isolation layer deforms elastically, the eigenvalue problem of the MDOF model is expressed as:

$$\left(\left[K_e\right]-{\omega }_e^2\left[M\right]\right)\left\{{\phi }_e\right\}=\left\{0\right\}$$

(13)

where $\left\{{\phi }_e\right\}$ is the elastic first mode shape of the MDOF model or the first mode shape of the MDOF model when the isolation layer deforms elastically, ${\omega }_e$ is the natural frequency of the MDOF model when the isolation layer deforms elastically, $\left[M\right]$ is the mass matrix of the MDOF model in which $m_b$ and $m_{u1}~m_{u10}$ are used, and $\left[K_e\right]$ is the stiffness matrix of the MDOF model in which $k_{be}$ and $k_{u1}~k_{u10}$ are used. The matrices $\left[M\right]$ and $\left[K_e\right]$ are given as follows:

$$\left[M\right]=\left[\begin{array}{ccc}{m}_b& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & m_{u10}\end{array}\right]$$

(14)

$$\left[K_e\right]=\left[\begin{array}{ccc}{k}_{be}+k_{u1}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & k_{u10}\end{array}\right]$$

(15)

On the other hand, when the isolation layer deforms plastically, the eigenvalue problem of the MDOF model can be expressed by:

$$\left(\left[K_p\right]-{\omega }_p^2\left[M\right]\right)\left\{{\phi }_p\right\}=\left\{0\right\}$$

(16)

where $\left\{{\phi }_p\right\}$ is the plastic first mode shape of the MDOF model or the first mode shape of the MDOF model when the isolation layer deforms plastically, ${\omega }_p$ is the natural frequency of the MDOF model when the isolation layer deforms plastically, and $\left[K_p\right]$ is the stiffness matrix of the MDOF model in which $k_{bp}$ and $k_{u1}~k_{u10}$ are used, which is given by:

$$\left[K_p\right]=\left[\begin{array}{ccc}{k}_{bp}+k_{u1}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & k_{u10}\end{array}\right]$$

(17)

$\left\{{\phi }_e\right\}$ and $\left\{{\phi }_p\right\}$ as obtained from Equations (13) and (16), respectively, are then normalized to satisfy:

$${\left\{{\phi }_e\right\}}^T\left[M\right]\left\{{\phi }_e\right\}={\left\{{\phi }_p\right\}}^T\left[M\right]\left\{{\phi }_p\right\}=1$$

(18)

Figure 8a,b show the normalized $\left\{{\phi }_e\right\}$ and $\left\{{\phi }_p\right\}$ obtained from Equation (18) considering the different values of the structural parameters. Note on the vertical axis notations, b refers to the isolation layer, and Gr. refers to the ground. As indicated in Figure 8a, $T_u$ has significant impacts on both $\left\{{\phi }_e\right\}$ and $\left\{{\phi }_p\right\}$ because it significantly influences the stiffness of the upper structure (Figure 2). As indicated in Figure 8b, $T_f$ has negligible impacts on $\left\{{\phi }_e\right\}$ and small impacts on $\left\{{\phi }_p\right\}$ because it slightly influences the initial and post-yield stiffness of the isolation layer (Figure 5b). As indicated in Figure 8c,d, ${\alpha }_{sy}$ and ${\delta }_{sy}$ have small impacts on $\left\{{\phi }_e\right\}$ but none on $\left\{{\phi }_p\right\}$ because they have influences on the initial stiffness but none on the post-yield stiffness of the isolation layer (Figure 5c,d). These findings are used for detailed discussions of the influences of the structural parameters on ES models in Section 3.4.

3.3. Modeling Method for ES Models

Based on the normalized $\left\{{\phi }_e\right\}$ and $\left\{{\phi }_p\right\}$ obtained from Section 3.2, the modeling method for the ES model is introduced in this section.

Figure 9a,b show the ES model and its shear force–deformation relationship ($Q^{es}\text{-}{\delta }^{es}$ relationship). According to Equation (18), the mass of the ES model $m^{es}$ in Figure 9a can be written as:

$$m^{es}={\left\{{\phi }_e\right\}}^T\left[M\right]\left\{{\phi }_e\right\}={\left\{{\phi }_p\right\}}^T\left[M\right]\left\{{\phi }_p\right\}=1$$

(19)

This indicates the mass of the ES model can be simplified into 1, while the mass of the CS model is the sum of the upper structure and base isolation-layer masses (Equations (1), (4), and (A1) in Appendix A).

Using the normalized $\left\{{\phi }_e\right\}$ and $\left\{{\phi }_p\right\}$, the initial and post-yield stiffness of the ES model, $k_e^{es}$ and $k_p^{es}$ (Figure 9b), are defined as follows:

$$k_e^{es}={\left\{{\phi }_e\right\}}^T\left[K_e\right]\left\{{\phi }_e\right\}$$

(20)

$$k_p^{es}={\left\{{\phi }_p\right\}}^T\left[K_p\right]\left\{{\phi }_p\right\}$$

(21)

Based on the normalized $\left\{{\phi }_e\right\}$ (the reason is presented in Appendix B), the yield shear force of the ES model $Q_y^{es}$ is described as:

$$Q_y^{es}=\left\{∆{\phi }_{eb},∆{\phi }_{eu1},\cdots ,∆{\phi }_{eu10}\right\}\left\{\begin{array}{c}{Q}_{by}\\ {\phi }_{eu1\left(by\right)}\\ \begin{array}{c}⋮\\ {\phi }_{eu10\left(by\right)}\end{array}\end{array}\right\}$$

(22)

Here, $∆{\phi }_{eb}$, $∆{\phi }_{eui}$, and ${\phi }_{eui\left(by\right)}$ are given by:

$$∆{\phi }_{eb}={\phi }_{eb}$$

(23)

$$∆{\phi }_{eui}=\left\{\begin{array}{c}{\phi }_{eu1}-{\phi }_{eb} \left(i=1\right)\\ {\phi }_{eui}-{\phi }_{eu\left(i-1\right)} \left(i=\mathrm{2,3},\dots ,10\right)\end{array}\right.$$

(24)

$${\phi }_{eui\left(by\right)}=Q_{by}·\frac{∆{\phi }_{eui}}{∆{\phi }_{eb}}\left(i=\mathrm{1,2},\dots ,10\right)$$

(25)

where ${\phi }_{eb}$ and ${\phi }_{eui}$ are the mode shapes of the isolation layer and the $i$-th story of the upper structure in normalized $\left\{{\phi }_e\right\}$.

Using $Q_y^{es}$ and $k_e^{es}$ as obtained from Equations (20) and (22), respectively, the yield deformation of the ES model ${\delta }_y^{es}$ can be written as:

$${\delta }_y^{es}=\frac{Q_y^{es}}{k_e^{es}}$$

(26)

As indicated in Equations (20)–(22) and (26), the shear force–deformation relationship of the ES model was obtained based on $\left\{{\phi }_e\right\}$ or $\left\{{\phi }_p\right\}$, which is influenced by the natural period of the upper structure (Equations (2), (15), and (17)). Whereas, the natural period of the upper structure does not influence the shear force–deformation relationship of the CS model (Equations (A2)–(A5) in Appendix A).

Note that the ES model is assumed to be an initial stiffness-proportional damping model. Hence, based on the Biggs method, the damping ratio of the ES model ${\xi }^{es}$ (Figure 9a) is expressed as [33]:

$${\xi }^{es}=\frac{{\xi }_b·W_b+\sum _{i=1}^{10}{\xi }_u·W_{ui}}{W_b+\sum _{i=1}^{10}{W}_{ui}}$$

(27)

where $W_b$ and $W_{ui}$ are the maximum potential energy of the isolation layer and the $i\text{-th}$ story of the upper structure, which are given based on normalized $\left\{{\phi }_p\right\}$ (the reason is presented in Appendix C) as follows [33]:

$$W_b=\frac{1}{2}·k_{bp}·{\phi }_{pb}^2$$

(28)

$$W_{ui}=\frac{1}{2}·k_{ui}·{\phi }_{pui}^2$$

(29)

Here, ${\phi }_{pb}$ and ${\phi }_{pui}$ are the mode shapes of the isolation layer and $i\text{-th}$ story of the upper structure in normalized $\left\{{\phi }_p\right\}$.

As Equation (27) indicates, the damping ratio of the ES model is relevant to that of the upper structure. This is in contrast to Equation (A6) in Appendix A, where the damping ratio of the CS model is not related to the upper structure.

Based on normalized $\left\{{\phi }_e\right\}$ (the reason is presented in Appendix D), the wind force of the ES model $F^{es}$ is expressed as:

$$F^{es}={\left\{{\phi }_e\right\}}^T\left\{F\right\}$$

(30)

where $\left\{F\right\}$ is the wind force vector composed of $F_b$ and $F_{u1}~F_{u10}$. This shows the wind force of the ES model is computed using the elastic first model, which is influenced by the four structural parameters ($T_u$, $T_f$, ${\alpha }_{sy}$, and ${\delta }_{sy}$) according to Equations (2), (5)–(8), and (15). Meanwhile, the wind force of the CS model is the sum of the wind forces of the upper structure and isolation layer and is irrelevant to these four structural parameters (Equation (A7) in Appendix A).

3.4. Influences of the Structural Parameters on ES Models

The influences of the structural parameters on the ES model obtained from Section 3.3 are investigated in this section, and the findings are used for detailed discussions on the accuracy verification for the isolation-layer energy dissipation in Section 4.

Figure 10a–d show the shear force–deformation of the ES model ($Q^{es}\text{-}{\delta }^{es}$ relationship) obtained from Equations (20)–(22) and (26) considering the different values of the structural parameters. As indicated in Figure 10a, $T_u$ influences both the elastic and plastic ranges because it influences $\left\{{\phi }_e\right\}$ and $\left\{{\phi }_p\right\}$ (Figure 8a), in contrast to its lack of influence on the $Q_b$-${\delta }_b$ relationship (Figure 5a). Note that the $Q_b$-${\delta }_b$ relationship is in accordance with the shear force–deformation relationship of the CS model (Equations (A2)–(A5) in Appendix A). As indicated in Figure 10b, $T_f$ has a negligible influence on the elastic range and a small influence on the plastic range due to its negligible impacts on $\left\{{\phi }_e\right\}$ and small impacts on $\left\{{\phi }_p\right\}$ (Figure 8b), respectively, which is similar to the $Q_b$-${\delta }_b$ relationship (Figure 5b). As indicated in Figure 10c,d, ${\alpha }_{sy}$ and ${\delta }_{sy}$ have small influences on the elastic range, which is different from the $Q_b$-${\delta }_b$ relationship (Figure 5c,d), but none on the plastic range, which is similar to the $Q_b$-${\delta }_b$ relationship (Figure 5c,d). This is because they have small impacts on $\left\{{\phi }_e\right\}$ and but none on $\left\{{\phi }_p\right\}$.

Figure 11a–d show the damping ratio of the ES model ${\xi }^{es}$ obtained from Equation (27) considering different values of the structural parameters. As shown in Figure 11a,b, $T_u$ and $T_f$ influence ${\xi }^{es}$ due to their impacts on $\left\{{\phi }_p\right\}$ (Figure 8a,b). In contrast, ${\alpha }_{sy}$ and ${\delta }_{sy}$ do not influence ${\xi }^{es}$ (Figure 11c,d) because they have no impact on $\left\{{\phi }_p\right\}$ (Figure 8c,d). Moreover, ${\xi }^{es}$ increases with an increasing ${\xi }_u$ according to Figure 11a–d.

Figure 12a,b show the wind forces of the ES model $F^{es}$ considering different values of $T_u$ ($T_f=5.0 \mathrm{s}$, ${\alpha }_{sy}=0.03$, ${\delta }_{sy}=2.5 \mathrm{cm}$, and Force 01) obtained from Equation (30). In Figure 12a, $F^{es}$ for $T_u=1.5 \mathrm{s}$ nearly equals that for $T_u=3.5 \mathrm{s}$ in the along-wind direction, which indicates $T_u$ has a negligible influence on $F^{es}$. Although $T_u$ influences $\left\{{\phi }_e\right\}$, the mode shapes of the lower stories (isolation layer and 1st~5th story) and upper stories (6th~10th story) vary adversely as $T_u$ increases (Figure 8a), which leads to the negligible influence of $T_u$ on $F^{es}$. The same can be said for the influences of the other structural parameters ($T_f$, ${\alpha }_{sy}$, and ${\delta }_{sy}$) on $F^{es}$. Moreover, Figure 12b also shows the negligible influence of $T_u$ on $F^{es}$ in the across-wind direction for to the same reason.

The influences of the structural parameters on the ES model are summarized in

Table 1. Influences of structural parameters on the ES model.

	${\mathit{m}}^{\mathit{e}\mathit{s}}$	${\mathit{k}}_{\mathit{e}}^{\mathit{e}\mathit{s}}$	${\mathit{k}}_{\mathit{p}}^{\mathit{e}\mathit{s}}$	${\mathit{Q}}_{\mathit{y}}^{\mathit{e}\mathit{s}}$	${\mathit{\delta }}_{\mathit{y}}^{\mathit{e}\mathit{s}}$	${\mathit{\xi }}^{\mathit{e}\mathit{s}}$	${\mathit{F}}^{\mathit{e}\mathit{s}}$
${\xi }_u$	×	×	×	×	×	○	×
$T_u$	×	○	○	○	○	○	○
$T_f$	×	○	○	○	○	○	○
${\alpha }_{sy}$	×	○	×	○	○	×	○
${\delta }_{sy}$	×	○	×	○	○	×	○

○ denotes having influence and × denotes having no influence.

. All five structural parameters have impacts on the ES model and therefore may influence the isolation-layer energy dissipation in the ES model. This differs to Table A1 in Appendix A, where two structural parameters (${\xi }_u$ and $T_u$) have no impact on the CS model.

4. Accuracy Verification for Isolation-Layer Energy Dissipation

In order to judge whether the proposed ES model (Section 3) can be used to estimate the isolation-layer energy dissipation in base-isolated tall buildings under strong winds, this section focuses on verifying the accuracy of calculating the isolation-layer energy dissipation using the ES model by comparing the isolation-layer energy dissipation in the ES model and MDOF model in the along- and across-wind directions. In addition, the isolation-layer energy dissipation in the CS model is also verified in this section for comparison with the ES model.

After performing time history analysis for the ES model, the isolation-layer energy dissipation in the ES model (per second [23]) ${\dot{W}}_b^{es}$ is expressed as:

$${\dot{W}}_b^{es}=\frac{1}{t_1-t_0}{\int }_{t_0}^{t_1}{Q}^{es}·{\dot{\delta }}^{es}dt$$

(31)

where $Q^{es}$ is the shear force of the ES model, ${\dot{\delta }}^{cs}$ is the derivative of ${\delta }^{cs}$ with respect to time, and $t_0=50 \mathrm{s}$ and $t_1=650 \mathrm{s}$ (the same as Equation (12)).

Figure 13a–d show the isolation-layer energy dissipation in the ES model ${\dot{W}}_b^{es}$ (Equation (31)), CS model ${\dot{W}}_b^{cs}$ (Equation (A8) in Appendix A), and MDOF model ${\dot{W}}_b$ (Equation (12)) with variations in $T_u$ and ${\xi }_u$ (Figure 13a), in $T_f$ and ${\xi }_u$ (Figure 13b), in ${\alpha }_{sy}$ and ${\xi }_u$ (Figure 13c), and in ${\delta }_{sy}$ and ${\xi }_u$ (Figure 13d). The accuracy of the ES and CS models is evaluated by comparing ${\dot{W}}_b^{es}$ and ${\dot{W}}_b^{cs}$ with ${\dot{W}}_b$, respectively.

As indicated in Figure 13a, ${\dot{W}}_b^{cs}$ has constant values and a low accuracy (errors greatly exceeding 20%) in the along- and across-wind directions because $T_u$ and ${\xi }_u$ have no impact on the CS model (Table A1 in Appendix A). On the other hand, ${\dot{W}}_b^{es}$ shows a much higher accuracy than ${\dot{W}}_b^{cs}$ in the along- and across-wind directions because $T_u$ and ${\xi }_u$ influence the ES model (Table 1). Although the errors in ${\dot{W}}_b^{es}$ are less than 20% in the along-wind direction, these errors can be ignored because the values of ${\dot{W}}_b^{es}$ and ${\dot{W}}_b$ in the along-wind direction are significantly small. In the across-wind direction, the errors in ${\dot{W}}_b^{es}$ are within 20% when ${\dot{W}}_b<800 \mathrm{kN}·\mathrm{cm}/\mathrm{s}$ and within 10% when ${\dot{W}}_b\ge 800 \mathrm{kN}·\mathrm{cm}/\mathrm{s}$, which means ${\dot{W}}_b^{es}$ has a good accuracy.

As indicated in Figure 13b, ${\dot{W}}_b^{cs}$ also has a low accuracy (errors greatly exceeding 20%) in the along- and across-wind directions. Despite $T_f$ influencing the CS model (Table A1 in Appendix A), the result shows it has a small impact on ${\dot{W}}_b^{cs}$. On the other hand, ${\dot{W}}_b^{es}$ also has a much higher accuracy than ${\dot{W}}_b^{cs}$ in the along- and across-wind directions. As explained in the preceding paragraph, the errors in ${\dot{W}}_b^{es}$ in the along-wind direction can be ignored.

As indicated in Figure 13c, ${\dot{W}}_b^{es}$ has a much higher accuracy than ${\dot{W}}_b^{cs}$ in the along- and across-wind directions. In the across-wind direction, the errors in ${\dot{W}}_b^{es}$ are approximately within 20%. However, in the large range of isolation-layer energy dissipation (${\dot{W}}_b\ge 800 \mathrm{kN}·\mathrm{cm}/\mathrm{s}$), the errors in ${\dot{W}}_b^{es}$ are within 10%.

As indicated in Figure 13d, the errors in ${\dot{W}}_b^{es}$ are approximately within 20% in the along- and across-wind directions. In the across-wind direction, it is observed that the error in ${\dot{W}}_b^{es}$ slightly decreases as ${\dot{W}}_b$ increases.

To sum up, the proposed ES model has a much higher accuracy of isolation-layer energy dissipation than the commonly used CS model in the along- and across-wind directions. The errors in the isolation-layer energy dissipation in the ES model are approximately less than 20%, which shows a good accuracy. Especially in the large range of isolation-layer energy dissipation (${\dot{W}}_b\ge 800 \mathrm{kN}·\mathrm{cm}/\mathrm{s}$ in the across-wind direction), they are approximately within 10%, indicating a good accuracy. These errors are mainly due to:

(1)

Partial parametric calculation of the ES model ($Q_y^{es}$, ${\xi }^{es}$, and $F^{es}$) is based on the assumption of only one mode shape ($\left\{{\phi }_e\right\}$ or $\left\{{\phi }_p\right\}$), as mentioned in Appendix B, Appendix C and Appendix D, but the isolation layer in fact shows elasto-plastic deformation under strong winds.

(2)

The influence of higher-order modes may lead to these errors.

As a supplementary, Figure 14 shows an example of the time history of the accumulated isolation-layer energy dissipation in the ES model $W_b^{es}$, CS model $W_b^{cs}$, and MDOF model $W_b$ in the across-wind direction (${\xi }_u=2\%$, $T_u=2.5 \mathrm{s}$, $T_f=5.0 \mathrm{s}$, ${\alpha }_{sy}=0.03$, ${\delta }_{sy}=2.5 \mathrm{cm}$, and Force 03). $W_b^{es}$ of the ES model is close to $W_b$ of the MDOF model during 600 s. This demonstrates the validity of the ES model.

5. Conclusions

This paper analyzed a theoretical shear-type multi-degree-of-freedom (MDOF) model idealized from a base-isolated tall buildings subjected to strong winds. The base isolation layer composed of rubber bearings and steel dampers was modeled as an elasto-plastic SDOF model, while the upper structure was modeled as an elastic 10-DOF model. Time history analyses revealed the influences of the five structural parameters ((i) damping ratio and (ii) natural period of the upper structure, (iii) isolated period, (iv) yield shear force coefficient, and (v) yield deformation of the steel damper) on the isolation-layer energy dissipation in the MDOF model. These findings contribute to the proposal of equivalent single-degree-of-freedom (ES) models to estimate the isolation-layer energy dissipation in base-isolated buildings under strong winds. The following conclusions are drawn from this study:

(1)

In the along-wind direction, the isolation-layer energy dissipation in the MDOF model is very small and negligibly influenced by the above five structural parameters because the wind forces contain mean components.

(2)

In the across-wind direction, three structural parameters (damping ratio and natural period of the upper structure and yield shear force coefficient of the steel damper) have significant impacts on the isolation-layer energy dissipation in the MDOF model, while the other two structural parameters (isolated period and yield deformation of the steel damper) have slight impacts.

(3)

Considering the above findings, these five structural parameters were used in devising the ES model. Hence, their influences on the ES model are captured. This greatly contradicts the common single-degree-of-freedom (CS) model, which is not influenced by the damping ratio and natural period of the upper structure.

(4)

Moreover, by considering the variation in the first mode shape of the MDOF model according to the elasto-plasticity of the base isolation layer, an appropriate first mode shape was selected for modeling the ES model. As a result, the ES model reproduces accurately the isolation-layer energy dissipation of the first mode of a base-isolated tall building against strong winds.

(5)

A comparison of the isolation-layer energy dissipation in the ES and CS models versus that in the MDOF model shows the ES model outperforms the CS model.

Based on the above conclusions, structural engineers can benefit from the proposed ES model. Using the ES model, they can accurately and efficiently estimate the isolation-layer energy dissipation in base-isolated tall buildings under strong winds.

Like any other method, the ES model has its limitations as its modeling is based on the assumption of the first mode. Hence, for very tall buildings where the influence of higher-order modes is dominant, it may have significant inaccuracies. Moreover, its modeling method is limited to base-isolated tall buildings equipped with rubber bearings (isolators) and steel dampers.

Also, directly comparing the hysteresis loops of the ES and MDOF models is impossible as the two models have different shear force–deformation relationship.

Despite the above limitations and advantages, the ES model can be used to estimate the energy dissipation in base-isolated tall buildings subjected to ground motions [34]. It can also be applied to different building configurations such as inter-story isolated tall buildings [35,36] subjected to strong winds or ground motion for time-saving analyses. Future work can address all the above limitations, e.g., extending the ES model to incorporate other types of isolators (e.g., [37]) and considering other wind effects (e.g., [38]).