Seismic fragility analysis of adjacent inelastic structures connected with viscous fluid dampers

Abstract

Many scholars have conducted vibration control analysis on adjacent structures, so far. However, most of the previous studies always focus on the elastic analysis of the structures which usually can’t meet the multi-level seismic targets of structural safety and using function, and often ignore the adverse effects of near-field earthquakes. In view of the limitations of elastic vibration control analysis, two-dimensional models of some adjacent inelastic reinforced concrete frame structures which are connected with the viscous fluid dampers (modeled by Maxwell model) were set up, and incremental dynamic analysis was conducted under 20 near-field and 20 far-field seismic waves to get the entire seismic response process. The seismic fragility curves of the adjacent structures which were controlled and uncontrolled were obtained. The applicability of the Maxwell damper optimum parameter expressions under different seismic performance levels was studied and the differences of seismic response of the adjacent structures under the far-field and near-field earthquakes were compared. Finally, through a large number of parametric analysis and based on the principle of minimum seismic fragility of the adjacent structures, the appropriate damping values of the Maxwell dampers which show preferable control effects under different seismic waves and seismic intensities were proposed.

Keywords

adjacent structures near-field earthquakes optimal damping seismic fragility analysis seismic response vibration control

Introduction

With the increasing development of urbanization, there have been a large number of smaller spacing adjacent buildings. Collision will occur in those buildings when they suffer from strong earthquakes and thus bring structural damage. Consequently, many scholars have proposed to install the control devices between the adjacent structures, using the relative vibration between the adjacent structures to absorb the seismic energy. Zhu et al. (2000) proposed a control method to reduce the seismic response using the interaction between the master–slave structures. They derived the general expressions of the optimal stiffness and damping of the passive dissipated devices under the stationary white noise excitation. They also analyzed the influence of the structural parameters on the control effects. Afterward, Zhu et al. (2003) simplified the adjacent structures into 2-single-degree-of-freedom (SDOF) systems and used Viogt viscoelastic damping model to represent the passive connection unit to derive the general expressions of optimal stiffness and damping of the control devices which suffered from the random ground excitation using Kuhn–Tucker optimization principle. Since then, Zhu et al. (2000), Zhu and Iemura (2001), Zhu and Liang (2005), Zhu and Xu (2005), and Zhu et al. (2006), respectively, proposed the analytical expressions of the optimum parameters of the Kelvin and Maxwell dampers between catamaran SDOF systems based on the principle of energy statistics. Xu and Zhang (2002) obtained a closed-form solution of seismic response for the adjacent structures which were connected by linear quadratic Gussian (LQG) controller. Through this closed-form solution, parametric study of the structures between the adjacent multi-degree-of-freedom (MDOF) systems was performed, and the favorable parameters which can control and reduce the maximum response of the adjacent structures were found. Bhaskararao and Jangid (2006a, 2006b) studied the control effect of the friction dampers which were used to connect the adjacent structures. The results showed that the friction dampers can get a good energy dissipation effect. Bhaskararao and Jangid (2007) simulated the base acceleration as a harmonic vibration and smooth Gaussian white noise excitation and studied the seismic response of two adjacent SDOF systems which were connected by viscous dampers. They derived the structural motion equations and obtained the relative displacement and absolute acceleration response of the adjacent structures. The results showed that when the viscous dampers had appropriate damping, they can reduce the seismic response between the adjacent structures. Basili and Angelis (2007a, 2007b) analyzed the control effects of the nonlinear hysteretic controller which was used to connect the adjacent structures. By simplifying the adjacent structures into SDOF systems, they obtained the optimum parameters of the hysteretic controllers prone to the Gaussian white noise and filtered white noise excitation using equivalent linearization method. Ok et al. (2008) performed the optimized control study on the adjacent structures which were connected by the hysteretic dampers based on multi-objective optimization and stochastic equivalent linearization method. Through a large number of nonlinear random vibration analysis, they studied the robustness of the optimized design by considering seismic randomness.

In summary, although vibration control of the adjacent structures has been widely studied, the seismic performance and fragility analysis of the adjacent structures are required under different exceeding probabilities and different intensities, especially under near-field earthquakes which have the characteristics of high energy and short duration, to meet multi-level seismic feature targets of structural safety and using function. In addition, several major earthquakes in recent years make people aware that the earthquakes bring not only huge economic losses but also serious social implications (Kasai and Maison, 1997; Naeim et al., 2000; Sezen et al., 2003; Wang, 2008), which makes the community and the owners have multi-level requirements to the buildings’ seismic performance: the designed structures are not only able to withstand collision and collapse under strong earthquakes but also can be ensured that the using function will not be lost during the earthquakes in order to achieve multi-level seismic objectives. So it needs to introduce the concept of performance-based seismic design. Therefore, to be earthquake-prone countries, it has important theoretical significance and practical value to carry out the research of performance-based vibration control of the adjacent structures.

In this article, the incremental dynamic analysis (IDA) was conducted on the adjacent reinforced concrete frame structures connected with the viscous fluid dampers (VFDs) which were modeled by the Maxwell models under near-field and far-field earthquakes. Based on the results of IDA, a after-treatment program was performed to get seismic fragility curves of the controlled and uncontrolled adjacent structures which, respectively, suffered from near-field and far-field earthquakes. From the perspective of performance evaluation, the control effects of the Maxwell damper optimum parameter theoretical expressions (Zhu and Xu, 2005b) under different seismic waves and different seismic intensities were explored, and based on the principle of minimum seismic fragility, the appropriate damping parameter values which can get preferable control effects for both the adjacent structures under each performance levels were proposed.

Modeling of adjacent structures

Modeling of the 2-MDOF systems

Figure 1 displays a structure–damper system composed of two adjacent structures of n₁ and n₂ (n₁ > n₂) stories, respectively, connected by the VFDs at each story. It is assumed that the two adjacent structures are symmetric with their symmetry planes coinciding with each other. The earthquake acceleration is considered to input in the direction of the symmetric planes. Both the structures are assumed to be excited by the same seismic acceleration and spatial variations in seismic waves are ignored. The story heights of both the structures are the same, with different total heights.

Figure 1.

Adjacent structures linked by dampers.

Theoretical expressions of optimum parameters for Maxwell dampers

A generalized SDOF system

In practical engineering applications, a generalized SDOF model is used to approximately simulate the dynamic response of Structure 1 or 2 under earthquakes. Based on the principle of virtual work that the internal force and inertia force of interlayer shear are equal to each other, the generalized SDOF system of Structures 1 and 2 can be formed. The equation of motion can be expressed as follows

m_{i}^{*} \overset{\cdot\cdot}{z} (t) + c_{i}^{*} \overset{\cdot}{z} (t) + k_{i}^{*} z (t) = - L_{i} \overset{\cdot\cdot}{u} (t)

(1)

where i = 1–2, $m_{i}^{*} = {ψ_{i}}^{T} M_{i} ψ_{i}$ , $c_{i}^{*} = {ψ_{i}}^{T} C_{i} ψ_{i}$ , $k_{i}^{*} = {ψ_{i}}^{T} K_{i} ψ_{i}$ , and $L_{i} = {ψ_{i}}^{T} M_{i} I_{i}$ . $ψ_{i}$ is the first vibration mode of Structure i. $M_{i}$ , $C_{i}$ , and $K_{i}$ are the mass, damping, and stiffness matrices of Structure i, respectively. $L_{i}$ is the generalized equivalent load factor. $z (t)$ , $\overset{\cdot}{z} (t)$ , and $\overset{\cdot\cdot}{z} (t)$ are the generalized displacement, velocity, and acceleration, respectively.

Optimum parameters of Maxwell dampers

As shown in Figure 2, the connecting VFD is represented by the Maxwell model, for which the spring stiffness and the damping coefficient at zero frequency are $c_{0}$ and k. The equations of motion for the two adjacent SDOF structures with Maxwell model–defined fluid damper can be written as

\begin{matrix} m_{1} {\overset{\cdot\cdot}{x}}_{1} (t) + c_{1} {\overset{\cdot}{x}}_{1} (t) + k_{1} x_{1} (t) + f_{Γ} (t) = - m_{1} α_{1} {\overset{\cdot\cdot}{u}}_{g} (t) \\ m_{2} {\overset{\cdot\cdot}{x}}_{2} (t) + c_{2} {\overset{\cdot}{x}}_{2} (t) + k_{2} x_{2} (t) - f_{Γ} (t) = - m_{2} α_{2} {\overset{\cdot\cdot}{u}}_{g} (t) \\ f_{Γ} (t) + λ {\overset{\cdot}{f}}_{Γ} (t) = c_{0} ({\overset{\cdot}{x}}_{1} (t) - {\overset{\cdot}{x}}_{2} (t)) \end{matrix}}

(2)

where $c_{i} = 2 ζ_{i} \sqrt{k_{i} m_{i}}$ , $k_{i} = m_{i} k_{i}^{*} / m_{i}^{*}$ , $α_{i} = L_{i} / m_{i}^{*}$ (i = 1 or 2), and $m_{i}^{*}$ and $k_{i}^{*}$ are the generalized mass and stiffness of Structure i, respectively; $m_{i}$ and $ζ_{i}$ are the total mass and the first modal damping ratio of Structure i, respectively (i = 1 or 2). $f_{Γ} (t)$ is the output force of the Maxwell damper at time t. $λ$ is the relaxation time of the Maxwell damper, and $λ = c_{0} / k$ .

Figure 2.

Two-SDOF structures linked by Maxwell model–defined VFD.

Zhu and Xu (2005b) derived the explicit expressions to obtain the optimum parameters of VFD between 2-SDOF adjacent structures. For different optimization objectives, the formulas of optimum parameters ( $ξ_{o p t}$ and $χ_{o p t}$ ) for fluid dampers in Zhu and Xu (2005b) are summarized as follows:

1. Optimization criterion 1: minimizing the time-averaged energy of Structure 1

when $β \leq 1$

\begin{matrix} ξ_{opt} = \frac{(1 - β^{2}) \sqrt{μ}}{\sqrt{- μ β^{4} + μ (3 + 4 μ) + (4 + 6 μ) β^{2}}} \\ χ_{opt} = \frac{(1 + 2 μ + β^{2}) \sqrt{μ}}{\sqrt{- μ β^{4} + μ (3 + 4 μ) + (4 + 6 μ) β^{2}}} \end{matrix}}

(3)

when $β > 1$

\begin{matrix} ξ_{opt} = \frac{(β^{2} - 1) \sqrt{μ}}{2 (1 + μ) \sqrt{μ + β^{2}}} \\ χ_{opt} = 0 \end{matrix}}

(4)

where $β = ω_{2} / ω_{1}$ is the frequency ratio of Structure 2 to 1; $μ = m_{1} / m_{2}$ is the total mass ratio of Structure 1 to 2.

The optimal values of the relaxation time $λ$ and damping coefficient $c_{0}$ at zero frequency are given, respectively, as

\begin{matrix} c_{0} = 2 ξ_{opt} m_{1} ω_{1} \\ λ = χ_{opt} / ω_{1} \end{matrix}}

(5)

2. Optimization criterion 2: minimizing the time-averaged energy of the two structures

Likewise, the frequency ratio is restricted to $β \leq 1$ . The expressions can also be applied to the case of $β > 1$ only requiring reciprocal exchange of the two structures. Correspondingly, the optimal damping ratio at zero frequency and optimal non-dimensional relaxation time (Zhu and Xu, 2005b) of the Maxwell model–defined damper can be expressed as follows:

when $μ \geq 1$

\begin{matrix} ξ_{opt} = \frac{\sqrt{(1 + μ^{2}) (μ^{2} + β^{2}) {(1 - β^{2})}^{2}}}{(1 + μ) \sqrt{μ ((8 - μ) β^{4} + μ (8 μ - 1) + 18 μ β^{2})}} \\ χ_{opt} = \frac{((μ - 2) β^{2} + μ (2 μ - 1)) \sqrt{(1 + μ^{2})}}{\sqrt{μ (μ^{2} + β^{2}) ((8 - μ) β^{4} + μ (8 μ - 1) + 18 μ β^{2})}} \end{matrix}}

(6)

when $μ < 1$ , $β^{2} < μ (2 μ - 1) / (2 - μ)$

\begin{matrix} ξ_{opt} = \frac{\sqrt{(1 + μ^{2}) (μ^{2} + β^{2}) {(1 - β^{2})}^{2}}}{(1 + μ) \sqrt{μ ((8 - μ) β^{4} + μ (8 μ - 1) + 18 μ β^{2})}} \\ χ_{opt} = \frac{((μ - 2) β^{2} + μ (2 μ - 1)) \sqrt{1 + μ^{2}}}{\sqrt{μ (μ^{2} + β^{2}) ((8 - μ) β^{4} + μ (8 μ - 1) + 18 μ β^{2})}} \end{matrix}}

(7)

when $μ < 1$ , $β^{2} \geq μ (2 μ - 1) / (2 - μ)$

\begin{matrix} ξ_{opt} = \frac{\sqrt{(1 + μ^{2}) {(1 - β^{2})}^{2}}}{2 (1 + μ) \sqrt{(1 + μ) (μ + β^{2})}} \\ χ_{opt} = 0 \end{matrix}}

(8)

Generally, unless there are special requirements on Structure 1, optimization criterion 2 is always adopted to do passive control analysis of the Maxwell dampers as its purpose is to control the overall seismic response of both the adjacent structures. Therefore, in the following analysis, optimization criterion 2 which minimizes the time-averaged energy of the two structures is used to do performance-based seismic vibration control analysis and study the control effects of the optimum parameters under different performance levels and different seismic waves.

Numerical examples

Model parameters

The established models are administration buildings located in Xi’an city. The models are the adjacent structures; while Structure 1 is a 10-story reinforced concrete frame structure, Structure 2 is a 6-story reinforced concrete frame structure. The structural arrangement is uniform. One specimen of the models for each structure was selected to establish two-dimensional models for clearance. The frames were designed to 8° fortification, site classification II, seismic design group 2. The basic wind pressure is 0.35 kN/m² and the basic snow pressure is 0.25 kN/m². The longitudinal reinforcement is HRB335 and the stirrup is HPB300. The concrete strength class of the beams, columns, and plates is C35. The section size of the beams is 300 mm × 800 mm, and the section size of the columns is 750 mm × 750 mm for Structure 1. The section size of the beams is 300 mm × 800 mm, and the section size of the columns is 800 mm × 800 mm for Structure 2. The slab thickness is 100 mm for all models. The story height of each story for the adjacent structures is 3.6 m. The calculation models and finite element models of the adjacent structures are shown in Figure 3. Figure 3 also displays the section numbers and the longitudinal reinforcement of the cross section of the bottom floor columns.

Figure 3.

Models of the adjacent structures: (a) calculation models of the adjacent structures and (b) finite element models of the adjacent structures.

OpenSees program was used to establish the two-dimensional models of the adjacent structures to do IDA. A suite of 20 near-field and 20 far-field ground motions, which are recorded on the soil condition, are selected from Pacific Earthquake Engineering Research (PEER) Strong Motion Database as the input to IDAs. The uniaxialMaterial Maxwell was selected to simulate the Maxwell damper. The beams, columns, and Maxwell dampers are simulated by the nonlinear fiber beam–column units related to the displacement. The first natural frequency of Structure 1 is approximately ω₁ = 8.418 rad/s. The first natural frequency of Structure 2 is approximately ω₁ = 17.5254 rad/s. The total mass of Structure 1 is 303.0705 ton and Structure 2 is 203.8021 ton.

The Maxwell dampers were assigned to each floor of the adjacent structures, totally six dampers. While the total number of the Maxwell dampers is equal to the story numbers of the lower building for the two adjacent structures. Here, the optimization objective is chosen as minimizing the total vibration energy of the tow structures. Using the explicit expressions of the Maxwell model–defined VFD between 2-SDOF systems (Zhu and Xu, 2005b), the total optimal relaxation time is 0.0408 s, and the total optimal damping coefficient is 1.4478 × 10⁶ N s/m (for each damper, damping is 2.413 × 10⁵ N s/m).

Determination of performance levels of structures

The maximum drift ratio was selected as the engineering demand parameters (EDPs) which can characterize the overall damage index of the structure. The limit states of the structure were divided into immediately occupation (IO), slightly damage (SD), life safety (LS), and collapse prevention (CP). Each state corresponds to the maximum performance objectives which are listed in Table 1 (FEMA 356, 2000).

Table 1.

Limits of each performance limit state.

Performance levels	IO	SD	LS	CP
Maximum drift ratio	0.005	0.01	0.02	0.04

IO: immediately occupation; SD: slightly damage; LS: life safety; CP: collapse prevention.

IDA

Choosing peak ground acceleration (PGA) as intensity measure (IM) and the maximum drift ratio as damage measure (DM), IDA was conducted on the adjacent structures which, respectively, suffered from 20 near-field and 20 far-field earthquakes. The IDA curves of the adjacent structures which were controlled and uncontrolled of both the structures are illustrated in Figures 4 and 5 under the near-field earthquakes, and Figures 6 and 7 under the far-field earthquakes.

Figure 4.

IDA curves for Structure 1 under the near-field earthquakes: (a) Structure 1 (uncontrol) and (b) Structure 1 (control).

Figure 5.

IDA curves for Structure 2 under the near-field earthquakes: (a) Structure 2 (uncontrol) and (b) Structure 2 (control).

Figure 6.

IDA curves for Structure 1 under the far-field earthquakes: (a) Structure 1 (uncontrol) and (b) Structure 1 (control).

Figure 7.

IDA curves for Structure 2 under the far-field earthquakes: (a) Structure 2 (uncontrol) and (b) Structure 2 (control).

Figures 4 to 7 show that the IDA curves under the near-field earthquakes are comparatively more divergence than the IDA curves under the far-field earthquakes. The hysteresis characteristic of the curves is more obvious as well.

Comparing the control effects through the IDA curves for different structures under different seismic waves, it can be seen that when Structure 1 suffered from near-field and far-field earthquakes, the control effects for both seismic waves are not obvious. Nevertheless, the control effects of Structure 2 are much better when PGA is lower than 1.0 g, especially when Structure 2 is subjected to far-field earthquakes.

In order to more intuitively compare the differences between the IDA curves under the near-field and far-field earthquakes and the control effects of the Maxwell dampers, Figure 8 displays the mean IDA curves of the adjacent structures, which are controlled and uncontrolled, suffering from the near-field and far-field earthquakes.

Figure 8.

Mean IDA curves of (a) Structure 1 and (b) Structure 2.

Figure 8(a) plots the IDA curves of Structure 1 under different seismic waves and different control conditions. It can be observed from Figure 8(a) that for the same PGA (when PGA > 0.5 g), the maximum story drift ratios of Structure 1 when suffering from the far-field earthquakes are much smaller than the response values which were obtained from the near-field earthquakes, even under controlled situation. It can also be seen from Figure 8(a) that when Structure 1 is under elastic state, the seismic responses are nearly the same for all kinds of situations. With the increase in PGA, the difference becomes bigger and bigger. The seismic response under the near-field earthquakes when Structure 1 was uncontrolled is the biggest, which means that the influence of near-field earthquakes is much larger than the far-field earthquakes.

Comparing the control effects of Structure 1, for the near-field earthquakes, when the PGA is lower than 1.1 g, the Maxwell dampers just play a little role. However, when the PGA exceeds 1.1 g, the Maxwell dampers can effectively control the structural response. Meanwhile, for the far-field earthquakes, except for low PGA (when PGA is lower than 0.3 g, the Maxwell damper can slightly control the structural response), the Maxwell dampers will not control the seismic response, but will dramatically enlarge the structural response. Consequently, when suffering from far-field earthquakes, using the optimization equations to calculate the optimum parameters of Maxwell dampers is not applicable for Structure 1.

Figure 8(b) shows the IDA curves of Structure 2 under different seismic waves and different control conditions. It can be observed from Figure 8(b) that for uncontrolled situation, the seismic responses of Structures 2 are close to each other when undergoing the near-field and far-field earthquakes. It means that the near-field earthquakes can cause little influence to the seismic response of Structure 2. However, when referred to the controlled seismic response of Structure 2, it can be seen from Figure 8(b) that the seismic response under the near-field earthquakes is much bigger than that under the far-field earthquakes, and with the increase in PGA, the difference between the two seismic oscillations becomes larger and larger.

Comparing the control effects of Structure 2, for far-field earthquakes, the Maxwell dampers can always show good control effects under different PGAs. Nevertheless, for near-field earthquakes, the Maxwell dampers have some control effects within the scope of PGA equal to 1.4 g. When the PGA exceeds this range, the Maxwell dampers will enlarge the structural response.

For a clearer comparison of control effects of the adjacent structures under different seismic waves and seismic intensities, Figures 9 and 10 display the top floor displacement time histories of adjacent structures under the PGA of 0.2 and 0.9 g for near-field and far-field seismic waves. For clearance, each graphical gives a partial enlarged view of the top floor displacement for the adjacent structures and the rest of the top floor displacements of other 19 seismic waves are omitted.

Figure 9.

Top floor displacement time histories for adjacent structures under the near-field earthquakes: (a) Structure 1 and (b) Structure 2.

Figure 10.

Top floor displacement time histories for adjacent structures under the far-field earthquakes: (a) Structure 1 and (b) Structure 2.

Figure 9 shows that under the near-field earthquakes for Structure 1, when PGA = 0.2 g, the Maxwell dampers have little control effects. When PGA = 0.9 g, the Maxwell dampers can effectively control the structure’s top floor displacement. For Structure 2, the Maxwell dampers have better control effects for both PGA = 0.2 g and PGA = 0.9 g.

Under the far-field earthquakes as displayed in Figure 10, it can be seen that the top floor displacement of Structure 1 can be effectively controlled under PGA = 0.2 g. However, when PGA = 0.9 g, the Maxwell dampers even enlarge the displacement. For Structure 2, the Maxwell dampers have better control effects for both PGA = 0.2 g and PGA = 0.9 g as well.

Comparing Figure 9 with Figure 10, it can be observed that the adjacent structures basically have the same maximum displacements under PGA = 0.2 g for both the near-field and far-field earthquakes. However, when PGA = 0.9 g, the top floor displacements for the structures subjected to near-field earthquakes are much larger than the far-field earthquakes. It again proves that the role of near-field earthquakes cannot be ignored.

Seismic fragility analysis

Seismic fragility analysis was conducted on the adjacent structures to acquire different probability of occurrence under different seismic waves and performance levels. The applicability of the optimum parameters’ expressions of Maxwell dampers (Zhu and Xu, 2005b) when experiencing big nonlinear state was studied. Figures 11 and 12 show the seismic fragility curves of the adjacent structures suffering from the near-field and far-field earthquakes, respectively.

Figure 11.

Seismic fragility curves for adjacent structures under the near-field earthquakes: (a) fragility curves of Structure 1 and (b) fragility curves of Structure 2.

Figure 12.

Seismic fragility curves for adjacent structures under the far-field earthquakes: (a) fragility curves of Structure 1 and (b) fragility curves of Structure 2.

Figure 11 shows the seismic fragility curves of the adjacent structures subject to the near-field earthquakes. It can be seen that the Maxwell dampers have little control effect on Structure 1 under different performance levels (for CP performance level, the control effects are the best). For Structure 2, the Maxwell dampers have better control effects under IO and SD performance levels but will enlarge the structural response under LS and CP performance levels.

Figure 12 shows the seismic fragility curves of the adjacent structures subject to the far-field earthquakes. It can be found that for Structure 1, the Maxwell dampers have no significant control effects for all kinds of performance levels and even enlarge structural response under some performance levels, such as SD and LS. In contrast, for Structure 2, the Maxwell dampers have excellent control effects under each performance level.

In order to better compare the differences of seismic response between the near-field and far-field earthquakes of different performance levels, Figures 13 and 14 show the comparison of seismic fragility of the adjacent structures between near-field and far-field earthquakes.

Figure 13.

Comparison for Structure 1 under the near-field and far-field earthquakes: (a) fragility curves of Structure 1 (uncontrol) and (b) fragility curves of Structure 1 (control).

Figure 14.

Comparison for Structure 2 under the near-field and far-field earthquakes: (a) fragility curves of Structure 2 (uncontrol) and (b) fragility curves of Structure 2 (control).

Figure 13 shows that no matter controlled or uncontrolled, the probability of exceedance for Structure 1 under the near-field earthquakes is bigger than under the far-field earthquakes, and with the increase in PGA, the difference between the near-field and far-field earthquakes becomes large.

Figure 14 shows that when the adjacent structures are uncontrolled, the probability of exceedance for Structure 2 is almost the same for the near-field and far-field earthquakes; even under IO and SD performance levels, the probability for the far-field earthquakes is slightly bigger than the near-field earthquakes. When the adjacent structures are controlled, the probability of exceedance for Structure 2 is increased with the increase in PGA; the near-field earthquakes are much bigger than the far-field earthquakes. Therefore, in the near-field earthquake zone, it should particularly pay attention to the high-energy characteristic of the seismic vibration for adjacent structures when connected with control devices.

Damper hysteresis curves

In order to further observe the difference in energy consumption for the Maxwell dampers under different seismic waves and different seismic intensities to preferably understand the control performance of the dampers, Figures 15 and 16 give the hysteresis curves of the Maxwell dampers. Due to space limitations, only one near-field and one far-field earthquakes were selected under PGA = 0.2 g (representing elastic state) and PGA = 0.9 g (representing elastic–plastic state) as the demonstration.

Figure 15.

Damper hysteresis curves for different seismic waves (PGA = 0.2 g): (a) near-field earthquake (0.2 g) and (b) far-field earthquake (0.2 g).

Figure 16.

Damper hysteresis curves for different seismic waves (PGA = 0.9 g) (PGA = 0.2 g): (a) near-field earthquake (0.9 g) and (b) far-field earthquake (0.9 g).

Figure 15 shows that when PGA equals 0.2 g, the damper hysteresis curves for both the near-field and far-field earthquakes are plump, especially for the far-field earthquakes. But the maximum stroke and the maximum output power of the damper for the near-field earthquakes are two times bigger than the far-field earthquakes, which indicate the high-energy characteristics of the near-field earthquakes.

Figure 16 shows that when PGA attains 0.9 g, the damper hysteresis curves for the near-field earthquakes are less plump than the far-field earthquakes except for the stroke range of ±0.1 m. When beyond this range, the damper has smaller energy consumption role because of the big plastic deformation. However, the maximum stroke for the near-field earthquakes is also two times bigger than the far-field earthquakes. The output force for both the seismic waves is nearly the same, which means that under rare earthquakes the force acting on the structures is no longer increasing but the deformation has continued to increase. Comparing the damper hysteresis curves of the two seismic oscillations, as a whole, the energy dissipation capacity of the dampers under the near-field earthquakes is not good as the far-field earthquakes, and the maximum stroke and output power are twice larger as the far-field earthquakes, which is caused by the characteristics of short duration and high energy of the near-field earthquakes.

Parametric analysis

As previously stated in section “Seismic fragility analysis,” employing the optimum parameter expressions (Zhu and Xu, 2005b) which were derived based on the hypothesis of elastic state to calculate the parameters of Maxwell dampers cannot get preferable control effects for all kinds of performance levels of the adjacent structures (even enlarge the seismic response at some performance levels). In order to explore more appropriate damper parameters at all performance levels for both the adjacent structures, in this article, a large number of IDAs under different damping parameters and different seismic waves (including far-field and near-field earthquakes) were performed. Based on the IDA, the seismic fragility curves of the adjacent structures under different damping parameters were obtained. Because of the less impact of stiffness coefficients of the Maxwell dampers on the control effectiveness, the seismic fragility of the adjacent structures under different damping coefficients was only studied, and based on the principle of minimum seismic fragility, the appropriate damping parameters were proposed.

Figures 17 and 18 give the seismic fragility curves of the adjacent structures under different damping parameters when suffering from the near-field earthquakes. Meanwhile, in order to make a comparison with the seismic response under the far-field earthquakes, Figures 19 and 20 also give the seismic fragility curves under different damping parameters under the far-field earthquakes.

Figure 17.

Fragility curves of Structure 1 under different damping parameters under the near-field earthquakes: (a) fragility curves of Structure 1 (IO), (b) fragility curves of Structure 1 (SD), (c) fragility curves of Structure 1 (LS), and (d) fragility curves of Structure 1 (CP).

Figure 18.

Fragility curves of Structure 2 under different damping parameters under the near-field earthquakes: (a) fragility curves of Structure 2 (IO), (b) fragility curves of Structure 2 (SD), (c) fragility curves of Structure 2 (LS), and (d) fragility curves of Structure 2 (CP).

Figure 19.

Fragility curves of Structure 1 under different damping parameters under the far-field earthquakes: (a) Structure 1—IO, (b) Structure 1—SD, (c) Structure 1—LS, and (d) Structure 1—CP.

Figure 20.

Fragility curves of Structure 2 under different damping parameters under the far-field earthquakes: (a) Structure 2—IO, (b) Structure 2—SD, (c) Structure 2—LS, and (d) Structure 2—CP.

When the structures suffered from the near-field earthquakes, it can be observed from Figures 17 and 18 that under IO, SD, and LS seismic performance levels, Structures 1 and 2 mainly have the optimal damping as 2.4 × 10⁵ N s/m for each Maxwell damper, which is just the calculated values referring to Zhu and Xu (2005b). Meanwhile, when the adjacent structures experience the CP limit state, the optimal damping for Structure 1 in Figure 17(d) is 2.0 × 10⁵ N s/m. However, for Structure 2, the optimal damping is 1.0 × 10⁸ N s/m which can be found in Figure 18(d).

From Figure 19, it can be found that when Structure 1 undergoes the three stages of IO, SD, and LS under the far-field earthquakes, if the damper damping parameter is set to be 1.0 × 10⁴ N s/m, the probability of exceedance of the structure is minimal. Then, when Structure 1 experiences CP stage, even increasing the damping to 1.0 × 10⁶ N s/m, the probability of exceedance is very close to the probability of the damping of 1.0 × 10⁴ and 1.0 × 10⁵ N s/m, and when the damping is set to be 2.4 × 10⁵ N s/m according to the literature (Zhu and Xu, 2005b), the probability is actually maximum.

For Structure 2, it can be seen from Figure 20 that when the structure experiences the seismic performance levels of IO, SD, and LS, if the damper damping parameter is set to be 1.5 × 10⁵ N s/m, the probability of exceedance of the structure is minimal. Then, when Structure 2 experiences CP stage, and the damping is set to be 2.4 × 10⁵ N s/m according to Zhu and Xu (2005b), the probability of exceedance is minimum, and when set to 1.5 × 10⁵ N s/m is the second small.

Thus, it can be seen that different performance levels and different seismic waves for both the adjacent structures have different optimal damping parameters which are not always the values calculated by the literature (Zhu and Xu, 2005b). In order to find the more appropriate damping optimum parameters which can simultaneously control the probability of exceedance of the two structures at all performance levels, based on the principal of making the total probability of exceedance of the two structures minimum, the optimal damping parameter values were derived. Figure 21 shows the mean total exceedance of the probability of the two structures under different performance levels and different seismic waves.

Figure 21.

Mean total exceedance of probability of the adjacent structures under different damping parameters: (a) IO, (b) SD, (c) LS, and (d) CP.

Figure 21 shows that for the near-field earthquakes, when the damping parameter is set to be 2.413 × 10⁵ N s/m according to Zhu and Xu (2005b), the mean total exceeding probability of the two structures under IO, SD, and LS performance levels can nearly be minimal. In addition, it is 2.0 × 10⁵ N s/m under CP performance level (when set to 2.413 × 10⁵ N s/m, the mean total exceeding probability is the biggest). Thus, it can be seen that for the near-field earthquakes, in general case, adopting the optimum parameter expressions to calculate the optimal damping coefficients is feasible, except for the rare earthquake which makes the structures on the brink of collapse.

For the far-field earthquakes, when the damping parameter is set to be 1.5 × 10⁵ N s/m, the mean total exceeding probability of the two structures can be relatively the smallest under IO, SD, and LS performance levels. In addition, it is 2.413 × 10⁵ N s/m under CP performance level. In general case, it suggests that the damping coefficient of the Maxwell dampers can be 1.5 × 10⁵ N s/m under the far-field earthquakes, even under CP performance level.

Conclusion

The IDA and seismic fragility analysis have been performed on some adjacent structures which connected with Maxwell dampers, respectively, suffered from the near-field and far-field earthquakes. It can be found that

For Structure 1, no matter controlled or uncontrolled, the seismic response under the near-field earthquakes is larger than under far-field earthquakes. For Structure 2 without control, the seismic response under the near-field earthquakes is nearly the same as under the far-field earthquakes. However, when Structure 2 is controlled, the seismic response under the near-field earthquakes is larger than under the far-field earthquakes. Therefore, not in all cases the structural response under near-field earthquakes is larger than the response under far-field earthquakes. However, if the adjacent structures, which located in near-field earthquake zone, are connected by the Maxwell dampers, the performance of the structures should be improved.

For near-field earthquakes, except for CP performance level, it is basically feasible to adopt the analytical expressions of the optimum parameters of the Maxwell dampers which were calculated under linear conditions to obtain the optimal damping. For far-field earthquakes, in order to get preferable control effects for the Maxwell dampers under all performance levels, it is suggested to recalculate the optimal damping using performance-based seismic design method.

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: National Nature Science Foundation of China (51408443), Science Foundation of Hubei Province Department of Education (Q20141504), and China Postdoctoral Science Foundation (2013M542024).

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