Influence of biaxial interaction of hysteretic restoring base forces on wind-induced inelastic response of base-isolated tall buildings

Abstract

This study addresses the influence of biaxial interaction of hysteretic restoring forces of base isolation system on wind-induced response of base-isolated tall buildings. Both buildings with and without eccentricity in center of resistance are considered. Response history analysis is carried out to characterize the coupled responses of a square-shaped base-isolated tall building. A comprehensive parameter study is presented which covers a wide range of yielding level, response ratio and correlation of alongwind and crosswind base displacements. The results demonstrate that the biaxial interaction leads to increase in low-frequency component and decrease in resonant component of lower inelastic base displacement. However, the increase of low-frequency component of base displacement does not affect the upper building response relative to base isolation system. As a result, the upper building response is reduced by the influence of biaxial interaction. The biaxal interaction also results in fast growth of time-varying mean alongwind base displacement. The increase of low-frequency component can be significant when the yielding level of higher response is significant and two translational base displacements are quite different in magnitude. The correlation of two translational base displacements enhances the influence of biaxial interaction. For the base-isolated building with eccentricity, the alongwind and crosswind base responses are closer in magnitudes thus are less influenced by the biaxial interaction.

Keywords

Base-isolated tall building inelastic response eccentricity Biaxial hysteretic restoring force alongwind response crosswind response

Introduction

Base isolation systems with nonlinear hysteretic restoring force characteristics can help to improve performance of tall buildings in terms of comfort of occupants, functionality, non-damage to acceleration-sensitive contents and non-structural elements under seismic loading. Studies on wind-induced response of base-isolated tall buildings have demonstrated that, while the linear elastic response is increased due to reduction of modal frequency as compared to fixed-base building, the yielding of base isolation system leads to additional hysteretic damping thus help to reduce the dynamic response (Kareem 1997; Liang et al. 2002; Katagiri et al. 2011, 2012 and 2014; Ogawa et al. 2016; Feng and Chen 2019a and b, 2021). On the other hand, the yielding leads to building drift in alongwind direction in terms of time-varying mean displacement (e.g., Feng and Chen 2019a and b, 2021). Buildings having eccentricities in centers of mass and resistance show three-dimensional (3D) coupled motions to wind excitation (e.g., Kareem 1985; Chen and Kareem 2005a and b). Tian and Chen (2022) carried out an comprehensive characterization of base-isolated tall buildings with eccentricity in the center of resistance. The results illustrated that the base isolation is more effective in reducing the inelastic response of buildings with eccentricity, and help to reduce the amplification effect of eccentricity.

The translational hysteretic restoring forces of base-isolation system in both alongwind and crosswind direction can be coupled and affected by translational displacements and velocities in both translational directions. A biaxial hysteretic restoring force model is often used to consider the biaxial interaction of translational hysteretic forces (Park et al. 1986; Wang and Wen 2000; Harvey and Gavin 2014). The torsional restoring force of base isolation system is often represented by a linear elastic model. This biaxial interaction will result in coupling of inelastic building responses even for buildings without eccentricity.

This study addresses the influence of biaxial interaction of hysteretic restoring forces of base isolation system on wind-induced building responses. Both buildings with and without eccentricity are investigated. A comprehensive parametric study through response time history analysis is performed to explore the effect of biaxial interaction on time-varying mean and fluctuating responses of a square-shaped tall building. The results of this study help in developing improved understanding of coupled responses of base-isolated tall buildings under strong wind loads.

Response analysis framework

Equations of motion

A multi-story base-isolated building with eccentricities of mass and resistance is considered. The coordinates of center of mass (C.M.) and center of resistance (C.R.) at $i$ -th floor ( $i = 1,2, \dots, N$ ) are denoted as ( $e_{m x i}, e_{m y i}),$ and $(e_{x i}$ , $e_{y i}$ ) as shown in Figure 1, where the origin O at each floor is on the vertical z axis of cartesian coordinate system. The $i$ -th floor has mass $m_{i}$ , and moment of inertia around C.M., $I_{i c} = m_{i} r_{i}^{2}$ , where $r_{i}$ is the radius of gyration. The story stiffnesses in three directions are denoted as $k_{x i}, k_{y i}$ and $k_{0 θ i}$ at C.R. The wind story forces and story torque acting at origin O are $P_{x i} (t), P_{y i} (t)$ and $P_{θ i} (t)$ in alongwind, crosswind and torsional directions, respectively. The $i$ -th story translational and rotational displacements at origin O are denoted as $u_{x i} (t), u_{y i} (t)$ and $θ_{i} (t)$ , respectively.

Figure 1.

Base-isolated building with eccentricities.

The C.M. and C.R. of the base slab are located at $(e_{m b x}, e_{m b y})$ and $(e_{b x}, e_{b y})$ , respectively. The mass and moment of inertia around C.M. are denoted as $m_{b}$ and $I_{o b} = m_{b} r_{b}^{2}$ , where $r_{b}$ is the radius of gyration.

The restoring forces and torque at the C.R. of base slab, $F_{s b x}^{'}$ , $F_{s b y}^{'}$ , and $M_{s b θ}^{'}$ , are given as

F_{s b x}^{'} = η_{b x} k_{b x} u_{b x}^{'} + (1 - η_{b x}) k_{b x} z_{x}^{'}

(1a)

F_{s b y}^{'} = η_{b y} k_{b y} u_{b y}^{'} + (1 - η_{b y}) k_{b y} z_{y}^{'}

(1b)

M_{s b θ}^{'} = k_{0 b θ} θ_{b}

(1c)

where

u_{b x}^{'}, u_{b y}^{'}

and

θ_{b}

are displacements at C.R.;

z_{x}^{'}

and

z_{y}^{'}

are hysteretic displacements;

k_{b x}, k_{b y}

and

k_{0 b θ}

are the stiffness at the C.R. of the base slab;

η_{b x}

and

η_{b x}

are second stiffness ratios. It should be mentioned that the torsional stiffness of the isolation layer is generally related to the stiffness in both translational directions. As a result, the torsional stiffness also has nonlinear properties. It is assumed that the restoring torque and torsional displacement are within the linear elastic range thus can be modeled in a linear elastic model.

When the translational hysteretic restoring forces are uncoupled, $z_{x}^{'}$ and $z_{y}^{'}$ are defined by the uniaxial Bouc-Wen model (Wen, 1976)

{\dot{z}}_{x}^{'} = {\dot{u}}_{b x}^{'} - z_{x}^{'} | {\dot{u}}_{b x}^{'} | {| z_{x}^{'} |}^{n - 1} [β_{x} + γ_{x} s g n ({\dot{u}}_{b x}^{'} z_{x}^{'})]

(2a)

{\dot{z}}_{y}^{'} = {\dot{u}}_{b y}^{'} - z_{y}^{'} | {\dot{u}}_{b y}^{'} | {| z_{y}^{'} |}^{n - 1} [β_{y} + γ_{y} s g n ({\dot{u}}_{b y}^{'} z_{y}^{'})]

(2b)

where

β_{x}, γ_{x}, β_{y}, γ_{y}

and

n

control the shape of hysteretic cycle; in this study,

β_{x} = γ_{x}, β_{y} = γ_{y}

are used; The yielding displacements in two directions are

Δ_{x} = {(β_{x} + γ_{x})}^{- 1 / n}

, and

Δ_{y} = {(β_{y} + γ_{y})}^{- 1 / n}

, respectively.

When the translational restoring forces are coupled, the following isotropic biaxial hysteretic model is used (Harvey and Gavin, 2014)

{\dot{z}}_{x}^{'} = {\dot{u}}_{b x}^{'} - z_{x}^{'} J; {\dot{z}}_{y}^{'} = {\dot{u}}_{b y}^{'} - z_{y}^{'} J

(3a)

J = {| \frac{{\dot{u}}_{b x}^{'}}{Δ_{x}} | | \frac{z_{x}^{'}}{Δ_{x}} | [β_{0} + γ_{0} s g n ({\dot{u}}_{b x}^{'} z_{x}^{'})] + | \frac{{\dot{u}}_{b y}^{'}}{Δ_{y}} | | \frac{z_{y}^{'}}{Δ_{y}} | [β_{0} + γ_{0} s g n ({\dot{u}}_{b y}^{'} z_{y}^{'})]} {[{(\frac{z_{x}^{'}}{Δ_{x}})}^{2} + {(\frac{z_{y}^{'}}{Δ_{y}})}^{2}]}^{\frac{n - 2}{2}}

(3b)

where

β_{0} = β_{x} Δ_{x}^{n} = β_{y} Δ_{y}^{n}, γ_{0} = γ_{x} Δ_{x}^{n} = γ_{y} Δ_{y}^{n}

The normalized accumulated dissipated hysteretic energy until time instant $t$ can be defined as (Lee and Hong, 2010)

E_{n x} = \frac{(1 - η_{b x})}{Δ_{x}^{2}} \int_{0}^{t} z_{x}^{'} {\dot{u}}_{b x}^{'} d t; E_{n y} = \frac{(1 - η_{b y})}{Δ_{y}^{2}} \int_{0}^{t} z_{y}^{'} {\dot{u}}_{b y}^{'} d t

(4)

The hysteretic restoring forces and base displacements at the origin of base slab, i.e., $F_{s b x}, F_{s b y}, M_{s b θ}$ , and $u_{b x}, u_{b y}, u_{b θ},$ are related as

F_{s b} = K_{b 1 η} u_{b} + K_{b 2 η} z^{'}

(5a)

F_{s b} = {F_{s b x}, F_{s b y}, M_{s b θ} / r_{b}}^{T}; u_{b} = {u_{b x}, u_{b y}, u_{b θ}}^{T}; u_{b}^{'} = {u_{b x}^{'}, u_{b y}^{'}, r_{b} θ_{b}}^{T} = e_{b} u_{b}; z^{'} = {z_{x}^{'}, z_{y}^{'}}^{T}; e_{b} = [\begin{array}{c} 1 & 0 & - e_{b y} / r_{b} \\ 0 & 1 & e_{b x} / r_{b} \\ 0 & 0 & 1 \end{array}]; K_{b 1 η} = [\begin{array}{c} η_{b x} k_{b x} & 0 & (- \frac{e_{b y}}{r_{b}}) η_{b x} k_{b x} \\ 0 & η_{b y} k_{b y} & (\frac{e_{b x}}{r_{b}}) η_{b y} k_{b y} \\ (- \frac{e_{b y}}{r_{b}}) η_{b x} k_{b x} & (\frac{e_{b x}}{r_{b}}) η_{b y} k_{b y} & {(\frac{e_{b y}}{r_{b}})}^{2} η_{b x} k_{b x} + {(\frac{e_{b x}}{r_{b}})}^{2} η_{b y} k_{b y} + \frac{k_{0 b θ}}{r_{b}^{2}} \end{array}]; K_{b 2 η} = [\begin{array}{c} ({1 - η}_{b x}) k_{b x} & 0 \\ 0 & ({1 - η}_{b y}) k_{b y} \\ (- \frac{e_{b y}}{r_{b}}) ({1 - η}_{b x}) k_{b x} & (\frac{e_{b x}}{r_{b}}) ({1 - η}_{b y}) k_{b y} \end{array}]

(5b)

It is obvious that the eccentricity of base slab leads to a hysteretic relation of torsional moment around the origin with translational and torsional displacements at the origin of base slab.

The equations of motion of the base-isolated building are

M R {\ddot{u}}_{b} + M \ddot{u} + C \dot{u} + K u = P (t)

(6a)

(M_{b} + R^{T} M R) {\ddot{u}}_{b} + R^{T} M \ddot{u} + C_{b} {\dot{u}}_{b} + K_{b 1 η} u_{b} + K_{b 2 η} z^{'} = R^{T} P (t)

(6b)

M = [\begin{array}{c} M_{x x} & 0 & M_{x θ} \\ 0 & M_{y y} & M_{y θ} \\ M_{x θ} & M_{y θ} & M_{θ θ} \end{array}]; K = [\begin{array}{c} K_{x x} & 0 & K_{x θ} \\ 0 & K_{y y} & K_{y θ} \\ K_{θ x} & K_{θ y} & K_{θ θ} \end{array}]; P (t) = [\begin{array}{c} P_{x} (t) \\ P_{y} (t) \\ P_{θ} (t) / r \end{array}]; M_{s s} = d i a g (m_{1}, \dots, m_{N}) (s = x, y); M_{θ θ} = d i a g (m_{θ θ 1}, \dots, m_{θ θ N}); M_{b} = [\begin{array}{c} m_{b} & 0 & - (e_{m b y} / r_{b}) m_{b} \\ 0 & m_{b} & (e_{m b x} / r_{b}) m_{b} \\ - (e_{m b y} / r_{b}) m_{b} & (e_{m b x} / r_{b}) m_{b} & m_{b θ} \end{array}] C_{b} = d i a g (c_{b x}, c_{b y}, c_{b θ}); R = [\begin{array}{c} r & 0 & 0 \\ 0 & r & 0 \\ 0 & 0 & r \end{array}]; r = {\begin{array}{c} 1 \\ ⋮ \\ 1 \end{array}}; P_{s} (t) = {\begin{array}{c} P_{s 1} (t), \dots, P_{s N} (t) \end{array}}^{T} (s = x, y, θ); P_{θ} (t) / r = {P_{θ 1} (t) / r_{1}, \dots, P_{θ N} (t) / r_{N}}^{T}

(6c)

where

u = {u_{x}^{T}, u_{y}^{T}, u_{θ}^{T}}^{T}

are the building displacements vector relative to the base;

M, C a n d K a r e

mass, damping and stiffness matrices of the upper building

;

m_{i}

is the mass of i-th story;

m_{θ θ i} = m_{i} (1 + e_{m x i}^{2} / r_{i}^{2} + e_{m y i}^{2} / r_{i}^{2})

;

M_{b}

and

C_{b}

are the mass and linear damping matrix of base isolation system;

m_{b θ} = m_{b} (1 + e_{m b x}^{2} / r_{b}^{2} + e_{m b y}^{2} / r_{b}^{2})

;

c_{b x}, c_{b y}

and

c_{b θ}

are damping coefficients of the base;

P (t)

is story wind forces vector;

R

reflects displacements of upper building resulted from unit displacement in base. The expressions of mass and stiffness matrices

M_{x θ} = M_{θ x}^{'}

M_{y θ} = M_{θ y}^{'}

K_{s s} (s = x, y, θ), K_{x θ} = K_{θ x}^{'}

and

K_{y θ} = K_{θ y}^{'}

are omitted here for the sake of brevity. The damping matrix of upper building is assumed to be proportional to stiffness matrix and is determined according to the first modal damping ratio of upper building. As being pointed by Ryan and Polanco (2008), the damping matrix as proportional to mass matrix can result in undesirable large damping ratio of the first mode of base-isolated building.

The upper building displacements relative to base is represented in first $3 S (S \leq N)$ modal displacements as

u (t) \approx \sum_{j = 1}^{3 S} ϕ_{j} q_{j} (t) = ϕ q (t)

(7)

Where

ϕ_{j} = {ϕ_{j x}^{T}, ϕ_{j y}^{T}, ϕ_{j θ}^{T}}^{T}

is the j-th mode shape of upper building;

ϕ = [ϕ_{1}, \dots, ϕ_{3 S}]

is the

3 S

mode shape matrix;

q (t) = {q_{1}, q_{2}, \dots, q_{3 S}}^{T}

is the generalized displacement vector; superscript T denote matrix transpose operator.

Accordingly, the equations of motion are reduced to

\bar{M} \ddot{\bar{q}} (t) + \bar{C} \dot{\bar{q}} (t) + {\bar{K}}_{1} \bar{q} (t) + {\bar{K}}_{2} z^{'} (t) = Q (t)

(8a)

\bar{M} = [\begin{array}{c} M_{b} + R^{T} M R & R^{T} M ϕ \\ ϕ^{T} M R & M^{*} \end{array}]; \bar{C} = [\begin{array}{c} C_{b} & 0 \\ 0 & C^{*} \end{array}]; {\bar{K}}_{1} = [\begin{array}{c} K_{b 1 η} & 0 \\ 0 & K^{*} \end{array}]; {\bar{K}}_{2} = [\begin{array}{c} K_{b 2 η} \\ 0 \end{array}]; \bar{q} (t) = {\begin{array}{c} u_{b} (t) \\ q (t) \end{array}}; Q (t) = {\begin{array}{c} R^{T} \\ ϕ^{T} \end{array}} P (t) = {\begin{array}{c} Q_{F} (t) \\ Q_{0} (t) \end{array}}

(8b)

where

M^{*} = ϕ^{T} M ϕ, C^{*} = ϕ^{T} C ϕ, K^{*} = ϕ^{T} K ϕ

are generalized mass, damping and stiffness matrices of the corresponding fixed-base building, respectively;

Q_{F} (t) = R^{T} P (t)

and

Q_{0} (t) = ϕ^{T} P (t)

are the base shear forces and torque, and the generalized forces of the upper building.

The equations of motion can be represented in state-space equations as

\dot{v} (t) = g (v (t)) + \bar{D} Q (t)

(9a)

g (v (t)) = [\begin{array}{c} \dot{\bar{q}} \\ - {\bar{M}}^{- 1} \bar{C} \dot{\bar{q}} - {\bar{M}}^{- 1} {\bar{K}}_{1} \bar{q} - {\bar{M}}^{- 1} {\bar{K}}_{2} z^{'} \\ {{\dot{u}}_{b x}^{'} - z_{x}^{'} J, {\dot{u}}_{b y}^{'} - z_{y}^{'} J}^{T} \end{array}]; v (t) = {\begin{array}{c} \bar{q} \\ \dot{\bar{q}} \\ z^{'} \end{array}}; \bar{D} = [\begin{array}{c} 0 \\ {\bar{M}}^{- 1} \\ 0 \end{array}]

(9b)

The yielding of base-isolation system causes base displacements drift attributed to the existence of mean (static) wind load until reaching the steady-state displacements. The equations of the time-varying displacements can be obtained by taking the expected values of both sides of equation (6) or equation (9). The steady-state static displacements are determined by setting the expected values of hysteretic displacements, $E [z^{'} (t)] = 0 .$ Accordingly, the steady-state displacements are calculated as

K \bar{u} = \bar{P}; K_{b 1 η} {\bar{u}}_{b} = R^{T} \bar{P}

(10)

and in modal coordinates we have

{\bar{K}}_{1} \bar{q} = \bar{Q}

(11)

where

\bar{P}

is mean (static) story forces; and

\bar{Q}

is mean value of

Q (t)

. Obviously, the steady-state displacements are determined by the static loads and post-yielding stiffness of the system (Baber 1984; Feng and Chen 2019a).

Wind loading model

The story wind force coefficients of i-th floor, coefficients of bending moments and base torque are defined as

C_{P_{s i}} (t) = \frac{P_{s i} (t)}{0.5 ρ U_{H}^{2} B H_{0}} (s = x, y); C_{P_{θ i}} (t) = \frac{P_{θ i} (t)}{0.5 ρ U_{H}^{2} B^{2} H_{0}}; C_{M_{s}} (t) = \frac{M_{s} (t)}{0.5 ρ U_{H}^{2} B H^{2}} (s = x, y); C_{M_{θ}} (t) = \frac{M_{θ} (t)}{0.5 ρ U_{H}^{2} B^{2} H}

(12)

where

P_{s i} (t) (s = x, y, θ)

are i-th story forces;

M_{s} (t) (s = x, y, θ)

are base bending moments or torque;

ρ = 1.225 k g / m^{3}

is air density;

B

is the building width;

H_{0}

is the story height;

H

is the building height;

U_{H}

is the mean wind speed at building top.

The mean (static) alongwind story force coefficient is defined as

{\bar{C}}_{P_{x i}} = {\bar{C}}_{D} {(\frac{z_{i}}{H})}^{2 α_{s}}

(13)

where

{\bar{C}}_{D}

is mean drag force coefficient and assumed to be constant over the building height, which is related to the base bending moment coefficient

{\bar{C}}_{M_{x}}

{\bar{C}}_{D} = 2 {\bar{C}}_{M_{x}} (α_{s} + 1)

;

α_{s}

is wind load profile coefficient and is 1/4 in this study;

{\bar{C}}_{M_{x}}

= 0.5937 and

α_{s}

=1/4 are determined from the wind tunnel data for a square-shaped tall building model (TPU Aerodynamic Database);

z_{i}

is the elevation of i-th floor above the ground.

The power spectral density functions (PSDs) and coherence models of alongwind, crosswind and torsional story forces on a square-shaped tall building have been developed in Tian and Chen (2022) based on wind tunnel data (TPU Aerodynamic Database). It is assumed that the PSDs of story force coefficients at different stories but same direction have the same shape

S_{C_{P_{s i}}} (f) = σ_{C_{P_{s i}}}^{2} S_{C_{P_{s 0}}} (f) (s = x, y, θ)

(14)

where

σ_{C_{P_{s i}}}

is the standard deviation (STD) of

C_{P_{s i}} (t)

; and

S_{C_{P_{s 0}}} (f)

is calculated from the PSD of base bending moment or torque

S_{C_{M_{s}}} (f)

S_{C_{P_{s 0}}} (f) = {(\frac{H}{H_{0}})}^{2} S_{C_{M_{s}}} (f) / \sum_{i = 1}^{N} \sum_{j = 1}^{N} σ_{C_{P_{s i}}} σ_{C_{P_{s j}}} C o h_{s i j} (f) (\frac{z_{i}}{H}) (\frac{z_{j}}{H}) (s = x, y);

(15a)

S_{C_{P_{θ 0}}} (f) = {(\frac{H}{H_{0}})}^{2} S_{C_{M_{θ}}} (f) / \sum_{i = 1}^{N} \sum_{j = 1}^{N} σ_{C_{P_{θ i}}} σ_{C_{P_{θ j}}} C o h_{θ i j} (f)

(15b)

where

C o h_{s i j} (f) (s = x, y, θ)

are the coherence functions between

i

-th and

j

-th story forces in

s

direction.

In this study, the PSDs of $C_{M_{x}} (t)$ and $C_{M_{y}} (t)$ are determined according to AIJ recommendation (AIJ, 2004) AIJ 2004 but with $σ_{{C_{M}}_{x}} = 0.1363$ and $σ_{C_{M_{y}}} = 0.1746$ . Since the PSD of $C_{M_{θ}} (t)$ is not defined in AIJ recommendations, a power spectral model (Kijewski and Kareem 1998) is fitted from wind tunnel data with

The STDs of alongwind, crosswind and torsional story force coefficients are calculated as follows, which are derived by fitting the wind tunnel data (TPU Aerodynamic Database):

σ_{C_{P_{x i}}} = 0.09 {(\frac{z_{i}}{H})}^{0.57} + 0.25

(16a)

σ_{C_{P_{y i}}} = 0.33 + 0.03 (\frac{z_{i}}{H}) + {(\frac{z_{i}}{H})}^{2} - 1.1 {(\frac{z_{i}}{H})}^{3}

(16b)

σ_{C_{P_{θ i}}} = 0.072

(16c)

The coherence functions of story forces, coherence of story crosswind force and torsional moment, and the PSD of base torque coefficient are presented in Appendix. The PSDs of base bending moment coefficients and base torque coefficient are shown in Figure 2.

Figure 2.

PSDs of base bending moment and torque coefficients.

Building model

A 50-story tall building with a square cross section is considered. The building height H = 200 m, width B = 40 m and building density is 192 kg/m³. The building has no eccentricity in C.M., but eccentricity in C.R. The story mass $m_{i} = 1.2288 \times 10^{6}$ kg and moment of inertia about origin $I_{i C} = m_{i} r_{i}^{2}; r_{i} = r = B / 2 / \sqrt{6} = 8.1650$ m. The eccentricity of C.R. at each floor level is same, i.e., $e_{x i} / r_{i} = e_{y i} / r_{i} = - 0.40$ .

The upper building without eccentricity (fixed-base) has fundamental frequencies $f_{0 x 1}$ = 0.210 Hz, $f_{0 y 1}$ = 0.252 Hz, and $f_{0 θ 1}$ = 0.300 Hz. The corresponding mode shapes follow linear functions along the building height, i.e., $ϕ_{0 x 1} (z_{i}) = ϕ_{0 y 1} (z_{i}) = ϕ_{0 θ 1} (z_{i}) = z_{i} / H$ . It is noted that torsional displacement is referred to as $u_{θ i} = r_{i} θ_{i}$ . The story stiffnesses of the shear building are determined from fundamental frequencies and mode shapes as

k_{s i} = \frac{{(2 π f_{0 s 1})}^{2}}{ϕ_{0 s 1} (z_{i}) - ϕ_{0 s 1} (z_{i - 1})} \sum_{j = i}^{N} m_{j} ϕ_{0 s 1} (z_{j}) (s = x, y, θ)

(17)

The first modal damping ratios of the fixed-base building without eccentricity are

ξ_{0 x 1} = ξ_{0 y 1} = ξ_{0 θ 1} =

1%. The yielding displacements in two translational directions of the base isolation system are

Δ_{x} = Δ_{y} = 0.025

m. In Bouc-Wen hysteretic restoring force model,

β_{s} = γ_{s} = 0.5 Δ_{s}^{n} (s = x, y)

and n = 6, which is close to a bilinear model. The eccentricity of C.R. of base isolation system is

e_{b x} / r_{b} = e_{b y} / r_{b} = - 0.40

, where

r_{b} = \sqrt{I_{o b} / m_{b}} = 8.1650

m. The eccentricity of C.R. of base isolation system is same as the upper building. Table 1 lists the dynamic properties of the base isolation system. The linear damping of base isolation system is not considered, where

f_{b s} (s = x, y, θ)

are the building frequencies in three directions when the upper building is modelled as a rigid body.

Table 1.

Dynamic properties of base isolation system.

Direction	$m_{b} (k g)$ or $I_{0 b} (k g \cdot m^{2})$	$k_{b s} (k N / m)$ or $k_{0 b θ} (k N \cdot m)$	$f_{b s} (H z)$	$η_{b s}$
$x$	4.08( $10^{5})$	5.53 $(10^{5})$	0.476	0.12
$y$	4.08 $(10^{5})$	8.25 $(10^{5})$	0.580	0.12
$θ$	2.72 $(10^{7})$	7.23 $(10^{10})$	0.669	1

It has been illustrated that the reduced 4DOFs model in each direction (including three modes of upper building) can accurately represent the modal properties of the first two modes (even first three modes) in each direction of base-isolated building. Tables 2 and 3 show the modal properties of base-isolated building with initial stiffness of isolation system without and with the eccentricity. The generalized mass corresponds to the mode shape with the largest displacement as unity. Figure 3 shows the coupled mode shapes of three fundamental modes of building with eccentricity. The eccentricity leads to coupled modal shapes. The modal frequencies and generalized mass with dominant alongwind displacements are changed by −8% and 30%, respectively. These are −7% and 36% for the modes with dominant crosswind displacements, 16% and 24% for the modes with dominant torsional displacements. The reduced building model with 4DOFs in each direction, i.e., a total of 12 DOFs, is used in the response analysis of this study.

Table 2.

Dynamic modal properties of base-isolated building without eccentricity (initial stiffness).

Direction	$x$			$y$			$θ$
Mode	1	2	3	1	2	3	1	2	3
$f_{s i}$ (Hz)	0.196	0.484	0.769	0.235	0.582	0.924	0.279	0.691	1.096
$M_{s i}$ kg)	2.89	0.84	1.21	2.86	0.85	1.20	2.93	0.84	1.23
$ξ_{s i}$ (%)	0.80	2.05	3.32	0.81	2.06	3.33	0.80	2.05	3.32
$u_{b s}$	0.105	−0.125	0.220	0.101	−0.121	0.212	0.890	−1.047	1.858

Table 3.

Dynamic modal properties of base-isolated building with eccentricity (initial stiffness).

Dominant direction	$x$			$y$			$θ$
Mode	1	2	3	1	2	3	1	2	3
$f_{i}$ (Hz)	0.180	0.445	0.706	0.219	0.543	0.862	0.325	0.806	1.279
$M_{i}$ kg)	3.75	1.09	1.56	3.90	1.15	1.66	3.66	1.04	1.52
$ξ_{i}$ (%)	0.80	2.05	3.31	0.81	2.06	3.32	0.80	2.05	3.31
$u_{b x}$	0.105	−0.125	0.220	0.052	−0.062	0.109	0.022	−0.029	0.054
$u_{b y}$	−0.039	0.047	−0.082	0.102	−0.122	0.218	−0.042	0.047	−0.095
$u_{b θ}$	−0.042	0.050	−0.089	0.035	−0.041	0.070	0.109	−0.129	0.223

Figure 3.

Coupled mode shapes of base-isolated building with eccentricity.

Influence of biaxial interactions of hysteretic forces

Time-varying mean displacement

Building response history analysis (RHA) under different mean wind speeds is carried out using 4^th-order Runge-Kutta method. The mean wind speed is along positive x-axis. The wind speed at building top varies from 20 to 60 m/s. The wind load time histories in three directions are simulated simultaneously using spectral representation method (Shinozuka and Jan 1972; Chen and Kareem 2005c) with consideration of coherence functions of story forces and wind loadings in three directions. For comparison purpose, both base-isolated buildings with and without eccentricity of C.R. are considered. The time step is 0.04 s and duration is 13 min for each sample, where the first 3 min was removed for eliminating transient effect. There is a total of 100 samples simulated for each wind speed and the response statistics are calculated by ensemble average. The responses of interest are building top displacement and acceleration as well as base displacement of base-isolation system. The upper building displacement is relative to base displacement, and the upper building acceleration is absolute acceleration relative to ground. The building base bending moments and torque have similar response characteristics as the building top displacement thus will not be discussed.

Figures 4 and 5 show a comparison of response time histories and restoring force-displacement relations at $U_{H}$ = 60 m/s under uniaxial and biaxial hysteretic restoring force models for the based-isolated building without and with eccentricity. Both alongwind and crosswind base displacement show inelastic behavior. There is a drift in the alongwind base displacement. Detailed analysis of response statistics will be presented in the latter part of this study.

Figure 4.

Time history samples of base and top displacements and restoring forces of base-isolated building without eccentricity ( $U_{H}$ = 60 m/s).

Figure 5.

Time history samples of base and top displacements and restoring forces of base-isolated building with eccentricity ( $U_{H}$ = 60 m/s).

Figure 6 displays the steady-steady alongwind base displacements at different mean wind speed. The eccentricity also leads to slight displacements in crosswind and torsional directions. The steady-state alongwind displacement is proportional to wind speed squared, and is not affected by the eccentricity and biaxial interaction. Figure 7(a) and (b) show the time-varying alongwind base displacement normalized by the corresponding steady-state mean value, which is estimated from ensemble average of 100 simulated time history samples. The time-varying mean displacement grows faster at higher wind speed due to larger fluctuating response and more frequent yielding, which corresponds to a higher hysteretic damping. The biaxial effect leads to fast growth in the time-varying mean alongwind displacement. The biaxial effect is more noticeable for building without eccentricity.

Figure 6.

The steady-state alongwind mean base displacement.

Figure 7.

Effect of biaxial interaction on normalized time-varying mean alongwind base displacement of base-isolated building.

Statistics of fluctuating response

The fluctuating building response with mean load is the same as that without mean load since the hysteretic loop with non-zero mean load and response is simply moved to a new position without changing its shape (e.g., Roberts and Spanos, 2003; Feng and Chen, 2019a). The response statistics of fluctuating response can be calculated from the response without consideration of the mean load.

Figures 8, 9, 10 show the STDs of responses at different wind speeds. Figures 11, 12, 13 are the response peak factors. Figure 14 portrays the ratio of STD of response with biaxial model over that with uniaxial model at different wind speed. It is evident that the eccentricity amplifies the alongwind response, while the crosswind and torsional responses are less affected. The alongwind base displacement shows weaker softening non-Gaussian feature but the crosswind base displacement has apparent softening non-Gaussian feature with increased peak factor at higher wind speeds. The torsional base displacement and building top relative displacements in three directions can be considered to be Gaussian random processes. The biaxial interaction leads to an increase in alongwind base displacement and decrease in alongwind top displacement and acceleration. The biaxial interaction has less influence on response peak factors.

Figure 8.

STDs of alongwind responses.

Figure 9.

STDs of crosswind responses.

Figure 10.

STDs of torsional responses.

Figure 11.

Peak factors of alongwind responses.

Figure 12.

Peak factors of crosswind responses.

Figure 13.

eak factors of torsional responses.

Figure 14.

Effect of biaxial interaction on responses of base-isolated building.

Table 4 summarizes the response STDs and correlation coefficients at

U_{H} = 60

m/s. In the case of building without eccentricity, the normalized accumulated hysteretic energy levels in alongwind and crosswind directions are 44 and 163, respectively, with the uniaxial model. The crosswind direction has a higher level of yielding. The biaxial interaction leads to 15% increase of alongwind base displacement. The response power spectral analysis shown in Figure 15 indicated that this increase is a result of 15% increase in low-frequency component with a frequency less than 0.1 Hz and 5% decrease in resonant component around the first modal frequency. The increase in the low-frequency alongwind base displacement does not affect upper building relative displacement and acceleration as the frequency is much lower than the modal frequency of the upper building. The STDs of alongwind top displacement and acceleration are reduced by 12% and 10% attributed to the decrease in resonant alongwind base displacement. The crosswind base displacement is almost not affected by biaxial interaction. The biaxial interaction has no influence on torsional response. The normalized accumulated hysteretic energy levels in alongwind and crosswind directions are 60 and 178, respectively, with the biaxial model.

Table 4.

STDs and correlation coefficients of responses for base-isolated building without eccentricity ( $U_{H}$ = 60 m/s).

Response		STD			Correlation coefficient
		$σ_{x}$	$σ_{y}$	${10 σ}_{θ}$	$ρ_{x y}$	$ρ_{x θ}$	$ρ_{y θ}$
				w/o eccentricity
Base displacement ( $m$ or rad)	Uniaxial	0.0392	0.0308	0.0123	0.02	0.01	−0.01
Base displacement ( $m$ or rad)	Biaxial	0.0452	0.0312	0.0123	0.01	0.01	−0.01
Top displacement ( $m$ or rad)	Uniaxial	0.113	0.132	0.109	0.01	0.00	−0.01
Top displacement ( $m$ or rad)	Biaxial	0.0994	0.125	0.109	0.01	0.00	−0.01
Top acceleration $(m / s^{2} o r r a d / s^{2})$	Uniaxial	0.185	0.366	0.400	0.01	0.00	0.00
Top acceleration $(m / s^{2} o r r a d / s^{2})$	Biaxial	0.167	0.344	0.400	0.00	0.00	0.00
			w/ eccentricity
Base displacement ( $m$ or rad)	Uniaxial	0.0554	0.0336	0.0119	−0.03	−0.17	0.42
Base displacement ( $m$ or rad)	Biaxial	0.0558	0.0346	0.0113	0.01	−0.18	0.42
Top displacement ( $m$ or rad)	Uniaxial	0.147	0.149	0.103	−0.04	−0.36	0.66
Top displacement ( $m$ or rad)	Biaxial	0.136	0.138	0.0980	−0.11	−0.40	0.66
Top acceleration $(m / s^{2} o r r a d / s^{2})$	Uniaxial	0.274	0.341	0.333	0.19	0.08	0.22
Top acceleration $(m / s^{2} o r r a d / s^{2})$	Biaxial	0.252	0.315	0.310	0.17	0.08	0.20

Figure 15.

Comparison of PSDs of building responses (w/o eccentricity).

In the case of base-isolated building with eccentricity, the alongwind response is amplified. The normalized accumulated hysteretic energy levels in alongwind and crosswind directions are 158 and 154, respectively, at $U_{H}$ = 60 m/s with the uniaxial force model. Both alongwind and crosswind responses have similar level of yielding, which leads to less influence of biaxial interaction on response STD and time-varying mean base displacement. The base displacement is almost not affected, while the building top displacements in two directions are reduced by 7%. The normalized accumulated hysteretic energy levels in alongwind and crosswind directions are 166 and 191, respectively, with the biaxial model.

Parameter study

Response STDs

The influence of biaxial interaction of hysteretic forces is affected by three parameters: yielding levels of alongwind and crosswind base displacements in terms of $σ_{u_{b x}} / Δ_{x}$ and $σ_{u_{b y}} / Δ_{y}$ , and their correlation coefficient $ρ_{x y}$ . In the following, the base-isolated building without eccentricity is used for a parameter study, but the fundamental frequencies of the building in alongwind and crosswind directions are set the same as 0.235 Hz. To eliminate the potential influence of response power spectral shape, the crosswind loading power spectrum at $U_{H}$ = 60 m/s is used to simulate both alongwind and crosswind loads but with different scaling parameters. Different yielding levels in two directions are achieved by scaling the simulated wind loading time histories. Different correlation coefficient of responses is achieved by changing the correlation of the loadings in two directions. As building in both directions have same modal frequency and damping ratio, the correlation coefficient of both responses will be identical to that of loadings when building responses are linear elastic.

Figure 16 shows the biaxial effects on building base and top displacements, which are given in terms of ratio of response STD with biaxial interaction to that without biaxial interaction, i.e., uniaxial case. In the uniaxial case, the hysteretic forces in alongwind and crosswind directions are uncoupled, thus the alongwind and crosswind responses are estimated separately. On the other hand, in the biaxial case both responses are calculated simultaneously. Three different levels of crosswind response $σ_{u_{b y}} / Δ_{y}$ = 0.51, 0.84 and 1.25 under uniaxial load, two correlation coefficients of loadings $ρ_{M_{x y}} = 0$ and 0.75, and different response ratio $r_{x y} = (σ_{u_{b x}} / Δ_{x}) / (σ_{u_{b y}} / Δ_{y})$ with $r_{x y} \leq 1$ , i.e., the alongwind response is less than crosswind, are considered. The normalized accumulated hysteretic energy levels of crosswind response at these three response levels are 13, 64, and 165, respectively.

Figure 16.

Biaxial effects on building response.

It is observed that the alongwind base displacement is increased, while the upper building alongwind displacement and acceleration are decreased by the biaxial interaction. The influence of biaxial interaction increases when both response levels are quite different and the larger response has a greater level of yielding. The increase in correlation of responses leads to greater influence of biaxial interaction. The upper building crosswind response is slightly decreased by biaxial interaction. The exception is when both alongwind and crosswind base displacements are well corelated, compatible and have high level of yielding, where the biaxial interaction results in considerable increase in both alongwind and crosswind base displacement.

Figure 17 shows the time history samples of alongwind and crosswind base and top displacements with uniaxial and biaxial models when $σ_{u_{b y}} / Δ_{y}$ = 1.25, $r_{x y}$ = 0.16 and $ρ_{M x y}$ = 0. Figure 18 displays the PSDs of responses. The biaxial interaction leads to increase in the low-frequency component and reduction in the resonant component of alongwind base displacement. Since the increase in low-frequency component is larger than the decrease in resonant component, the alongwind base displacement increases by 23% under the influence of biaxial interaction. The increase in low-frequency component of alongwind base displacement does not affect the relative response of upper building, since the low frequencies are much lower than the resonant frequency of the upper building. Thus, the biaxial interaction only leads to decrease in resonant components of alongwind top displacement and acceleration. The crosswind response is greater than the alongwind response and is not affected by biaxial interaction.

Figure 17.

Response time histories ( $σ_{u_{b y}} / Δ_{y} = 1.25, r_{x y} = 0.16, ρ_{M_{x y}} = 0$ ).

Figure 18.

PSDs of building responses ( $σ_{u_{b y}} / Δ_{y} = 1.25, r_{x y} = 0.16, ρ_{M_{x y}} = 0$ ).

When alongwind and crosswind base displacements are comparable ( $r_{x y} \approx 1$ ), the biaxial interaction has little influence on the STDs of base displacements when $ρ_{M x y}$ = 0, but leads to increase in both alongwind and crosswind base displacements when $ρ_{M x y}$ = 0.75, especially when the yielding levels are high.

Response correlation

The biaxial interaction also influences the correlation of alongwind and crosswind displacements. When the alongwind and crosswind base displacements are independent under the uniaxial model, the biaxial interaction cannot increase their correlation. However, when they have correlation under uniaxial model, the biaxial interaction can improve their correlation as shown Figure 19. Under uniaxial model, the correlation of base displacements is much lower than the correlation of wind loadings in two directions, especially at higher level of yielding. Under biaxial load, the correlation of base displacements is close to the correlation of loadings, and is not affected by the level of yielding. The correlation of alongwind and crosswind building top responses is different from that of the base displacements. This difference is related to their different PSD characteristics.

Figure 19.

Correlation coefficients of alongwind and crosswind responses ( $ρ_{M x y} = 0.75)$

The correlation of building responses is generally lower than that of wind loadings. For the linear elastic response, the correlations of alongwind and crosswind responses is determined by the correlation/coherence of wind loadings, modal frequencies and damping ratios (Chen and Kareem 2005a and b). The building responses can be independent when building modal frequencies are well separated even the wind loadings are fully correlated.

Time-varying mean responses

Figure 20 shows the time-varying mean alongwind base displacements with uniaxial and biaxial models for $σ_{u_{b x}} / Δ_{x} = 0.20 (σ_{u_{b y}} / Δ_{y} = 0.51, r_{x y} = 0.40; σ_{u_{b y}} / Δ_{y} = 1.25, r_{x y} = 0.16)$ and $σ_{u_{b x}} / Δ_{x} = 0.40 (σ_{u_{b y}} / Δ_{y} = 0.51, r_{x y} = 0.80;$ $σ_{u_{b y}} / Δ_{y} = 1.25, r_{x y} = 0.32)$ . It is observed that the biaxial interaction leads to fast growth of time-varying mean of alongwind base displacement. The correlation between the alongwind and crosswind base displacement has almost no influence on the alongwind time-varying mean base displacement. The steady-state alongwind displacement is not affected by biaxial effect.

Figure 20.

Time-varying mean of alongwind base displacement.

Influence of the shape of loading power spectrum

To investigate the influence of the shape of loading PSD on biaxial interaction, the wind loads simulated from alongwind loading PSD when $U_{H}$ = 60 m/s are exerted on building in both alongwind and crosswind directions instead. Figure 21 shows the biaxial effects of base and top displacements for different parameters. The normalized accumulated hysteretic energy levels of crosswind response at these three response levels are 3, 10, and 25. With the use of the alongwind loading PSD, the biaxial interaction leads to larger increase in low-frequency component but less decrease in resonant component of alongwind base displacement. Thus, the biaxial interaction leads to less reduction in alongwind top displacement and acceleration.

Figure 21.

Biaxial effects on building responses (Use of alongwind loading PSD).

When the alongwind and crosswind base displacements are comparable ( $r_{x y} \approx 1$ ), the biaxial interaction also has little influence on the STDs of base displacements when $ρ_{M x y}$ = 0, but leads to increase in both alongwind and crosswind base displacements when $ρ_{M x y}$ = 0.75.

Conclusions

The influence of biaxial interaction of translational hysteretic restoring forces of base-isolation system on response of base-isolated buildings depends on the yielding levels of alongwind and crosswind base displacements, their correlations and power spectral characteristics. The lower response of a direction is more affected by the biaxial interaction, while the higher response of another direction is less influenced. The biaxial interaction leads to increase in low-frequency component and decrease in resonant component of base displacement in the direction with lower response. At higher yielding level, the increase in low-frequency component is greater than the decrease in resonant component. As a result, the base displacement is increased due to biaxial interaction. However, the increase of low-frequency component of base displacement does not affect the upper building relative response. Thus, the biaxial interaction leads to decrease in upper building relative responses in the direction with lower response.

The influence of biaxial interaction increases when both response levels are quite different and are strongly correlated and the larger response has greater level of yielding. When both alongwind and crosswind base displacements are well correlated, comparable and have significant level of yielding, the biaxial interaction leads to increase in low-frequency components of both alongwind and crosswind base displacements. Thus, both base displacements are increased by the influence of biaxial interaction. The well-correlated responses are more affected and become more correlated due to the biaxal interaction. The influence of biaxial interaction is greater when the base displacements have higher low-frequency components. It should be mentioned that when both alongwind and crosswind base displacements are within linear elastic, they do not have interaction thus can be estimated separately.

The biaxial interaction leads to fast growth of time-varying mean alongwind base displacement. The correlation between alongwind and crosswind base displacement has no influence on the alongwind time-varying mean base displacement. The steady-state alongwind displacement is not affected by biaxial interaction. The modelling of biaxial interaction of hysteretic forces is more important for quantification of base displacement as compared to upper building relative response.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The support for this work provided in part by National Science Foundation (NSF) grant No. CMMI-2153189 is greatly acknowledged.

ORCID iD

Xinzhong Chen

Appendix

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