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Abstract

The present study investigates the performance of a semi-active spring (SAS) in the mitigation of the seismic response of base-isolated structures. Initially, under stationary filtered white-noise earthquake excitation, the response of the multi-floor flexible base-isolated structure with SAS is examined to observe the response control effects. The equivalent linearization technique is used to obtain the stochastic response, as the force-deformation behaviour of SAS is non-linear. The performance of SAS in terms of added stiffness, damping, and response mitigation to the isolated structure is also investigated. It is found that the SAS controls the isolator displacement effectively. Furthermore, it was noted that there is an optimum stiffness value for the SAS devices for a particular system and excitation, at which point the RMS top floor acceleration reaches a minimum value. Next, approximate formulas are proposed for the RMS isolator displacement, the top floor absolute acceleration, and the optimum stiffness of the SAS. It is observed that these formulas accurately predict the expected response and can be applied to the initial design of base-isolated structures using SAS. Finally, using the non-linear model of the SAS, the seismic response of flexible base-isolated structures is determined for actual near-fault earthquakes, considering different values of the isolation periods and stiffness of the SAS device. The SAS was effective in controlling the isolator displacement under near-fault motions. The trends of the results of isolated structures with SAS devices under near-fault earthquake motions were also in good agreement with those under stochastic excitation.

Keywords

Equivalent linearization empirical equations near-fault earthquake seismic base isolation semi-active spring stochastic excitation

Introduction

Seismic base isolation is a useful technique for safeguarding structures and their contents during strong earthquakes. Over the past 40 years, various types of base isolation devices have been developed to mitigate the harm that earthquakes cause to civil engineering structures. The new construction and strengthening of important buildings, such as barracks, hospitals, fire stations, and emergency management headquarters, have made full use of some of the isolation systems.¹ One of the major concerns in base-isolated structures is reducing the isolator displacement to ensure their safety and stability during strong seismic events. To control the isolator displacement, supplemental devices such as viscous, visco-elastic, friction, magnetorheological, negative stiffness, shape memory alloys, tuned mass dampers, tuned mass damper-inerter, tuned inerter damper, fluid inerter damper, electromagnetic inertial mass damper, clutched inerter damper, and negative stiffness inerter-based damper, along with the base isolation system, have been proposed and studied.^2–13 These supplemental devices are found to be effective in controlling the seismic response of base-isolated structures. Apart from studying supplementary control devices for isolated structures, the researchers have also explored new vibration absorbers and advanced seismic analysis technologies.^14–18

In addition to the supplementary devices mentioned above, semi-active control devices in particular have also drawn a lot of attention in the past because of their advantages, which include low power requirements, maintaining the adaptability of active control systems, and providing reliability comparable to passive control devices. As an alternative to passive systems, hybrid systems that combine semi-active systems with base-isolated structures have been put forward and investigated.^19–22 Typical semi-active devices that are taken into consideration in these circumstances are magnetorheological and fluid viscous dampers.^23–26 Oliveira et al.²⁷ presented a comparative analytical analysis of various control strategies for semi-active devices installed in buildings isolated from the base with the goal of minimizing vibrations caused by earthquakes. Through experiments, Gu et al.²⁸ have demonstrated that the magnetorheological elastomer base isolation system can effectively suppress the structural responses, and the general regression neural network inverse model can accurately reproduce the desired control force. Based on numerical simulations, it is demonstrated that the adaptive fractional-order fuzzy proportional-integral-derivative controller is more effective in reducing the seismic responses of a structure that is isolated from its base and is excited by a variety of real-data earthquakes.²⁹ A semi-active electromagnetic friction damper is capable of reducing the seismic response of multi-story base-isolated buildings.³⁰ The distributed tunable friction pendulum system can mitigate the displacement of the isolation layer without significantly increasing floor acceleration.³¹ The multi-objective modified clipped optimal controller and passive-on mode represent fewer failure probabilities of structures under earthquakes than the uncontrolled and passive-off modes.³² The transmissibility-based semi-active controller adds the requisite damping to structures under long as well as short-period ground motion.³³ A semi-active tuned mass damper with variable stiffness and damping can improve the displacement and acceleration performances of base-isolated structures.³⁴ A variable-orifice damper for a smart base-isolation system results in optimal control of structures under earthquakes with different frequency characteristics.³⁵ Li et al.³⁶ have shown that the semi-active control method using frequency-dependent variable damping can provide optimal control of the structure under various types of ground motions. An excellent review of the development and current knowledge of semi-active devices for base-isolated structures was recently presented by Sheikh et al.³⁷ The above review of research work indicates the popularity of semi-active devices for base-isolated structures. However, the performance of a semi-active spring (SAS) as a supplemental device for the base-isolated structures has not yet been studied.

Herein, the seismic response and behaviour of the base-isolated structure with a supplemental SAS device are investigated. The specific objectives of the study are summarized as follows: (i) investigate the effectiveness of SAS in controlling the seismic response of the base-isolated structures subjected to stationary earthquake excitation; (ii) compare the performance of the SAS device with the corresponding simple spring for the base-isolated structures; (iii) propose empirical equations for the approximate response and optimal stiffness of the SAS device for the base-isolated structures; and (iv) investigate the effectiveness of the SAS devices for base-isolated buildings under near-fault earthquakes.

Base-isolated structure with SAS

An N-floor shear-type building with base isolation and SAS is chosen as a structural model, as shown in Figure 1. The following assumptions are made for the structural system under consideration: (i) the selected building is symmetric with no torsional effects and no impact on the adjacent structures,³⁸ (ii) each story of the superstructure is assumed to have rigid floors with linear force-deformation behaviour and viscous damping, and (iii) the isolation system is modelled by the stiffness, k_b, and damping, c_b representing the equivalent stiffness and damping of the isolation system, respectively. The two parameters can be used to characterize the isolation system in the selected structural model, namely the isolation period (T_b) and the isolation damping ratio ( $ξ_{b}$ ) defined as

T_{b} = 2 π \sqrt{\frac{M}{k_{b}}}, and 2 ξ_{b} ω_{b} = \frac{c_{b}}{M}

(1)

where

M = (m_{b} + \sum_{i = 1}^{N} m_{i})

is the isolated structure’s total mass; and m_i is the mass of the ith floor; m_b is the mass of the base raft or isolation floor; N is the number of floors or stories in the superstructure; and ω_b = 2π/T_b is the isolation frequency.

Figure 1.

Structural model of the flexible base-isolated structure with supplemental SAS.

Mathematical modelling of SAS

The SAS applies the required resisting force to the structure when it moves away from its initial position (i.e. displacement, $x_{b}$ and velocity, ${\dot{x}}_{b}$ moving in the same direction). This can be accomplished by creating a spring that only produces force during stretching or by using a semi-active actuator to deliver the required control force.³⁹ Thus, the resisting force, F_s of the SAS is expressed as

F_{s} = {\begin{cases} k_{s} x_{b} \\ 0 \end{cases} \begin{array}{l} when (x_{b} {\dot{x}}_{b}) > 0 \\ when (x_{b} {\dot{x}}_{b}) \leq 0 \end{array}

(2)

The stiffness of the SAS is normalized to the isolator stiffness by

α = \frac{k_{s}}{k_{b}}

(3)

Governing equations of motion

The chosen base-isolated building’s governing equations of motion are

M \ddot{x} (t) + C \dot{x} (t) + Kx (t) + D F_{s} (t) = - Mr {\ddot{x}}_{g} (t)

(4)

where

M

represents the mass matrix;

C

denotes the damping matrix;

K

stands for the stiffness matrix; x(t) = {x_b, x₁, x₂, ….., x_N}^T implies the vector containing the relative horizontal displacements of the isolation and superstructure floors at time t; x_b designates the horizontal displacement of the isolation floor; x_i represents the horizontal displacement of the ith superstructure floor; T implies the transpose;

D

denotes the location matrix;

F_{s} (t)

represents the vector of control forces produced by the SAS device;

r = {1, 1, 1, 1, 1, 1}^{T}

symbolizes the influence coefficient vector; and

{\ddot{x}}_{g} (t)

represents the earthquake acceleration.

The mass of each floor, including the isolation floor, is kept constant, as is the stiffness of all floors, which is expressed by the parameter k = k_i (i = 1 to N). The value of k is chosen so that the fundamental period of the superstructure as a fixed base is N/10. The damping matrix of the superstructure is not known explicitly, and it is built by assuming a modal damping ratio of 2% in all modes of vibration. With the above-assumed values of parameters, the model of the base-isolated structure with the SAS considered in the present study requires the specifications of N, T_b, ξ_b, and α.

Response under filtered white-noise earthquake excitation

Unpredictable and multi-dimensional ground motions are always present during earthquakes. If the frequency content evolution is disregarded, the ground motion can be represented by a power spectral density function (PSDF). The Kanai-Tajimi^40,41 model of the stationary PSDF is considered in the present study and defined as

S_{{\ddot{x}}_{g}} (ω) = S_{0} (\frac{1 + 4 ξ_{g}^{2} {(ω / ω_{g})}^{2}}{{[1 - {(ω / ω_{g})}^{2}]}^{2} + 4 ξ_{g}^{2} {(ω / ω_{g})}^{2}})

(5)

where

S_{{\ddot{x}}_{g}} (ω)

is the PSDF of the earthquake acceleration

{\ddot{x}}_{g}

; S₀ is the amplitude of the power spectrum of the white-noise process; and

ω

_g and ξ_g are the predominant frequency and damping ratio of the soil media or filter, respectively. The values of the parameters selected are

S_{0}

= 0.05 m²/sec³, ω_g = 15 rad/sec, and

ξ_{g}

= 0.6 which corresponds to a firm type of soil media.

Let the model of the flexible base-isolated structure with SAS considered in Figure 1 be excited by an earthquake having the power spectrum as specified by Equation (5). Considering that Equation (2) illustrates the non-linear characteristics of the resisting force of the SAS, it needs to be converted into the equivalent linear system for the spectral stochastic response analysis. The equivalent linearized model is used to replace the non-linear behaviour of the SAS, expressed as

F_{s} = k_{e q} x_{b} + c_{e q} {\dot{x}}_{b}

(6)

where

k_{e q}

and

c_{e q}

are the equivalent constants for the SAS.

According to the procedure developed by Roberts and Spanos,⁴² the values of the equivalent constants $k_{e q}$ and $c_{e q}$ are evaluated by minimizing the expected value of the difference between the linear and non-linear counterparts of the force of the SAS device. The values of the equivalent constants are expressed in Ref. [39] as

k_{e q} = \frac{k_{s}}{2} and c_{e q} = \frac{k_{s}}{π} \frac{σ_{x_{b}}}{σ_{{\dot{x}}_{b}}};

(7)

where the terms

σ_{x_{b}}

and

σ_{{\dot{x}}_{b}}

in the expression of the equivalent damping stand for the root mean square (RMS) relative displacement and velocity of the base mass of the isolated structure, respectively. It will be worth mentioning here that the non-linear phenomenon of SAS still exists because the equivalent damping constant is dependent on the response of the isolated structure.

Let represent the harmonic acceleration of an earthquake at time t, where $ω$ is the circular frequency, and $j = \sqrt{- 1}$ . The mean square of the selected response quantity, $σ_{R}^{2}$ can be expressed as

σ_{R}^{2} = {\int_{- \infty}^{\infty} | H_{R} (j ω) |}^{2} S_{{\ddot{x}}_{g}} (ω) d ω

(8)

where

H_{R} (j ω)

is the frequency response function (FRF) of the desired response quantity.

The stationary mean square displacement and velocity of the base mass of the isolated structure with SAS are determined by solving Equation (8) in an iterative manner. After each iteration, the equivalent constant $c_{e q}$ is modified and re-evaluated against the response until full convergence is achieved. It was determined that 10 iterations were sufficient, and the results converged to the fourth decimal place.

To assess the performance of the SAS for the base-isolated structures, the FRF and RMS responses were calculated for various response quantities. The selected response quantities of interest are: (i) the relative displacement in the isolation system (x_b), (ii) the top floor absolute acceleration (i.e. ${\ddot{x}}_{N}^{a} = {\ddot{x}}_{N} + {\ddot{x}}_{g}$ ), (iii) the force in the isolator (F_iso), (iv) the force in the SAS (F_sas), and (v) the total base shear (F). The forces are normalized with the total weight of the isolated structure, W = Mg (i.e. g is the gravitational acceleration).

The FRF of top floor absolute acceleration (i.e. $H_{{\ddot{x}}_{N}^{a}} (j ω)$ ) and bearing displacement (i.e. $H_{x_{b}} (j ω)$ ) of the base-isolated structure with SAS and a simple spring (SS) are plotted in Figure 2. These FRF are plotted for one- and five-floor isolated structures with SAS considering the parameters as T_b = 3 s, $ξ_{b}$ = 0.1, and $α$ = 0.3. The figure demonstrates that the FRF amplitudes of absolute acceleration of the highest floor and displacement of the isolation system are decreased at isolation frequency because of the SAS. The figures also show that the higher frequencies do not contribute to the FRF amplitude of the displacement of the isolator, which has a peak primarily associated with the area around the isolation frequency. However, at higher frequencies, the SAS increases the FRF of absolute acceleration at the highest floor. This implies that the optimal stiffness of the SAS should be used to achieve the maximum possible reduction in structural accelerations. In addition, the responses are also plotted in Figure 2 for the simple spring (SS) having the same stiffness as that of the SAS for comparison purposes. The SS increases the acceleration of the highest floor in comparison to that of the isolation system. However, the displacement of the isolation system decreases with SS, but it is still higher than that of the SAS. This implies that the SAS is a better supplemental device for base-isolated structures, which reduces floor accelerations as well as isolator displacement in comparison to the SS device.

Figure 2.

Variation of FRF of top floor absolute acceleration and bearing displacement of a base-isolated structure with supplemental SAS (T_b = 3 s, ξ_b = 0.1, and $α$ = 0.3).

The values of the equivalent damping constant of the SAS (i.e. $c_{e q}$ ) are plotted in Figure 3 along with the corresponding iteration numbers to assess the convergence phenomenon. The constant $c_{e q}$ is initially set to zero and updated after the analysis has computed the $σ_{x_{b}}$ and $σ_{{\dot{x}}_{b}}$ . The updated value of $c_{e q}$ is applied at the start and updated at the end of the subsequent iteration. This continues until the values of $c_{e q}$ at the start and end of the iteration are equal. Figure 3 demonstrates that after 10 iterations, there is good convergence to the equivalent damping constant of the SAS, and the same was considered for the evaluation of the response of the base-isolated structure with SAS.

Figure 3.

Variation of the equivalent damping constant, $c_{e q}$ against the number of iterations (T_b = 3 s, and ξ_b = 0.1).

Figure 4 depicts the effects of the stiffness of the SAS on RMS top floor acceleration in one- and five-story structures for various isolation periods and damping ratios. The figure shows that as the stiffness of SAS increases, the RMS top floor acceleration decreases at first and reaches a minimum value before increasing as the stiffness of SAS increases further. This indicates that the SAS has an optimum stiffness for which the top floor acceleration of the superstructure is at its lowest. Further, the optimum value of the stiffness of the SAS decreases with an increase in the isolation period or flexibility and damping of the isolation system. When the optimal value of the stiffness of the SAS device is compared with the one- and five-story isolated structures, it is lower for the structures with a higher number of floors. It will be worth noting here that the effects of the stiffness of the SAS on the structural accelerations of the isolated structure are quite similar to the effects of viscous damping in elastomeric bearings.³⁷

Figure 4.

Influence of the stiffness of the supplemental SAS on the RMS top floor acceleration of the base-isolated structures.

Figure 5 depicts the effects of the stiffness of the SAS on the RMS bearing displacement in one- and five-story structures for various isolation periods and damping ratios. It is observed that the bearing displacement decreases with the increase in stiffness of the SAS for all values of the isolation periods and damping ratios considered.

Figure 5.

Influence of the stiffness of the supplemental SAS on the RMS bearing displacement of the base-isolated structures.

From Figures 4 and 5, the RMS top floor acceleration of the base-isolated structures with supplemental SAS decreases as the isolation period gets longer. Bearing displacement, on the other hand, increases as the isolation period elongates. Thus, the earthquake forces transmitted to the structure can be reduced while the bearings’ relative displacement increases. The relative base displacement, on the other hand, has a practical limit. As a result, a compromise between transmitted earthquake forces and relative bearing displacements must be made when designing the seismic isolation system.

Figure 6 depicts the effects of the stiffness of SAS on the variation of RMS force in the isolation system, the force in SAS, and the entire base shear. With an increase in the stiffness of the SAS, the force in the isolation system decreases, and the force in the SAS increases, as expected. The force in the SAS is greater than that of the isolation system at higher values of the stiffness of the SAS. The entire base shear initially decreases with the increase in stiffness of the SAS, and after that, it remains almost constant.

Figure 6.

Influence of the stiffness of the supplemental SAS on the RMS force in the isolation system, force in the SAS device, and entire base shear of the base-isolated structure.

Approximate response and optimum stiffness of SAS

It is difficult to find the exact solution to the integral Equation (8) for the mean square response of the flexible base-isolated structure with the supplemental SAS subjected to filtered white-noise excitation. However, some approximate closed-form expressions can be presented by considering (a) the superstructure as rigid in comparison to the isolation system, and its flexibility is disregarded, though the superstructure flexibility has some effects on the seismic performance of isolated buildings,⁴³ and (b) the earthquake acceleration in the vicinity of the isolation frequency as local white-noise idealization. The later consideration implies that the amplitude of the power spectrum of the local white-noise process will be $S_{{\ddot{x}}_{g}} (ω_{b})$ . The $S_{{\ddot{x}}_{g}} (ω_{b})$ is the value of the PSDF from equation (5) at the isolation frequency, $ω_{b}$ . In view of the above, the following expressions for the mean square displacement, velocity, and absolute acceleration of the base-isolated structure with SAS are hereby proposed:

σ_{x_{b}}^{2} = \frac{π S_{{\ddot{x}}_{g}} (ω_{b})}{2 ξ_{b} ω_{b}^{3} (1 + α_{c}) (1 + α_{k})}

(9)

σ_{{\dot{x}}_{b}}^{2} = \frac{π S_{{\ddot{x}}_{g}} (ω_{b})}{2 ξ_{b} ω_{b} (1 + α_{c})}

(10)

σ_{{\ddot{x}}_{a}}^{2} = π S_{{\ddot{x}}_{g}} (ω_{b}) ω_{b} [\frac{1}{2 ξ_{b}} (\frac{1 + α_{k}}{1 + α_{c}}) + 2 ξ_{b} (1 + α_{c})]

(11)

where

α_{k}

and

α_{c}

are the ratios defined as

α_{k} = \frac{k_{e q}}{k_{b}} and α_{c} = \frac{c_{e q}}{c_{b}}

(12)

Figure 7 shows a comparison of the RMS top floor absolute acceleration and bearing displacement of the base-isolated structures with the SAS as determined by stochastic analysis versus that predicted by the proposed approximate equation (i.e. refer to Equations (9) and (11)). It has been discovered that the responses to the two approaches are very similar and accurate for a one-story base-isolated structure. However, there is some difference in the results predicted by the two approaches for the five-story base-isolated structure. Hence, the proposed equations for the mean square displacement and absolute acceleration of the base-isolated structure predict the expected response with reasonable accuracy, and they can be utilized in the preliminary design of the base-isolated structures with supplemental SAS.

Figure 7.

Comparison of RMS top floor absolute acceleration and bearing displacement of the base-isolated structures with supplemental SAS by the approximate closed-form expressions (T_b = 3 s and ξ_b = 0.1).

As observed in Figure 4, there exists an optimum value of the stiffness of the SAS device for which the RMS absolute acceleration attains the minimum values. This can be obtained by differentiating the equation (11) and setting it to zero for the minimum value. By following this, the expression for the optimum values of the stiffness of the SAS will be

ξ_{b} (\frac{1 + α_{c}}{\sqrt{1 + α_{k}}}) = \frac{1}{2}

(13)

When there is no supplemental device in the isolation system, the absolute acceleration in equation (13) becomes the minimum when

ξ_{b} = \frac{1}{2}

. This is consistent with the value of the optimum damping in linear isolation as reported in the reference.⁴⁴

Flexible isolated structure under near-fault earthquakes

For a specified time history of the earthquake ground motion, the equations (4) are numerically integrated to determine the seismic response of the isolated building with the SAS device.⁴⁵ The non-linear behaviour of the SAS was duly considered in the time history analysis, and a 10⁻⁴ sec time interval is used for numerical integration such that the transition from the engaged to disengage condition of the SAS device and vice versa just takes place at the end of the time interval. To assess the efficiency of the SAS device as a potential supplemental damper, earthquake records from near-fault (NF) are chosen. 4-time histories of the actual NF recorded earthquakes are considered, as mentioned in Table 1.

Table 1.

Details of the selected NF earthquake ground motions.

Earthquake	Recording station	Year	PGA(g)	Component
Northridge	Sylmar Olive	1994	0.843	EW
Imperial Valley	El Centro Array #5	1979	0.383	EW
Kobe (Japan)	Takarazuka	1995	0.697	NS
Loma Prieta	LGPC	1989	0.570	NS

The time variation of the highest floor total acceleration, isolator displacement, the force in the isolation system, the force in the supplemental SAS device, and the entire base shear are depicted in Figure 8. The response is shown for the Northridge (Sylmar Olive), 1994 NF earthquake ground motion, and compared with the corresponding response of an isolated structure without supplemental SAS (referred to as BIS). The figure shows that the peak value of the highest floor acceleration for isolated structures with the SAS device is slightly increased in comparison to that of BIS, implying that these supplemental devices are increasing the highest floor acceleration. However, the increase in acceleration is insignificant when compared to the corresponding peak-highest floor acceleration of the fixed base structure. Additionally, it has been found that for isolated structures with the SAS device, the highest floor acceleration is associated with high-frequency amplitudes. These structural accelerations could severely affect certain types of equipment that are sensitive to high frequencies. There is a reduction in isolator displacement by the SAS device. It is observed to have decreased by 13%, respectively, as compared to the corresponding BIS. The variation of force in the isolation system is similar to that of the isolator displacement. The peak value of the force in the SAS device for the isolated structure is almost equal to the $α$ times of the corresponding isolator force. When compared to the corresponding entire base shear of the BIS, the peak value of the entire base shear for isolated structures equipped with the SAS device is marginally increased.

Figure 8.

Time history of the highest floor total acceleration, isolator displacement, force in the isolation system, force in the SAS, and entire base shear of a five-story base-isolated building subjected to the Northridge (Sylmar Olive), 1994 earthquake (T_b = 3 s, ξ_b = 0.1, and $α$ = 0.5).

Figure 9 displays the plots of the corresponding resisting forces in the isolation system and SAS device against the isolator displacement. The force, as well as displacement, are reduced in the isolation system because of the SAS device. In addition, the force-deformation behaviour of the SAS is taken into account in Equation (2) is also supported by the plot.

Figure 9.

The plot of the restoring forces of the isolation system and SAS against the isolator displacement of a five-story base-isolated building subjected to the Northridge (Sylmar Olive), 1994 earthquake (T_b = 3 s, ξ_b = 0.1, and $α$ = 0.5).

The effects of the stiffness ratio α on the peak-highest floor total acceleration and isolator displacement of the base-isolated structures with the SAS are shown in Figure 10. The figure indicates that the peak absolute acceleration increases marginally for the structure with SAS. On the other hand, the peak bearing displacement of base-isolated structures with SAS decreases with an increase in α. By comparing the acceleration and isolator displacement responses of the base isolated structure equipped with supplemental SAS, it can be concluded that the SAS device with a stiffness ratio up to 0.5 is effective in controlling the isolator displacement with a decrease in the absolute acceleration or its marginal increase.

Figure 10.

Effect of the stiffness of the SAS on the peak-highest floor absolute acceleration and bearing displacement of the five-floor isolated structure under various earthquakes (ξ_b = 0.1).

Figure 11 shows the variation of the associated peak force in the isolator, the force in the SAS, and the entire base shear of the base-isolated structure against the stiffness ratio α. As expected, an increase in the values of a causes the isolator force to decrease and the force in the SAS to increase. The entire base shear increases as the α rises. However, the increase in the entire base shear is substantial for the Northridge 1994 earthquake excitation compared to the other NF motions.

Figure 11.

Effect of the stiffness of the SAS on the peak force in the isolation system, force in the supplemental devices, and entire base shear of the five-floor isolated structure under NF earthquakes (T_b = 3 s and ξ_b = 0.1).

A comparison of the effects of the stiffness of the SAS in response control of isolated structures under stochastic earthquake excitation with that under recorded NF earthquake motions indicates the same trends in the variation of the highest floor absolute acceleration, the bearing displacement, the force in the isolator, the force in the SAS, and the entire base shear (refer to Figures 4–6 and Figures 10 and 11). The pattern of the isolated structure’s isolator displacement when supplemented with the SAS under actual near-fault earthquake records and the outcomes from stochastic earthquake excitation were found to be highly correlated.

Comparison of the linear and non-linear response

The non-linear analysis of the base-isolated structures subjected to earthquake time history is quite cumbersome, involving the engaged and disengaged conditions of the SAS device. To overcome this, it is explored to use the equivalent linear model of the SAS device. The equivalent linear model used here is the same as that proposed by equation (6) and the associated values of the equivalent constants given by equation (7). The equivalent stiffness coefficient is a constant quantity (i.e. equal to half of the stiffness of the SAS); however, the equivalent damping coefficient is dependent on the statistical response of the isolated structure. When the superstructure of the isolated structure is relatively rigid and subjected to broad-band earthquake excitation, the statistical responses can be approximated using equations (10) and (11) and will provide the value of the equivalent damping constant as

c_{e q} = \frac{k_{s}}{π ω_{b}} \sqrt{\frac{2}{2 + α}}

(14)

The peak seismic response of a five-story base-isolated structure with non-linear analysis is compared with the corresponding equivalent linear analysis by considering the equivalent stiffness and the damping of the SAS (refer to Equations (7) and (14) for equivalent stiffness and damping, respectively). Table 2 shows the comparative results for different stiffness ratios and four NF earthquake excitations. It is observed from the table that the difference in the highest floor total acceleration, isolator displacement, the force in the isolation system, and the entire base shear predicted by the two approaches is not quite significant for different values of the α. However, there is a difference in the force in the supplemental SAS device between the two approaches. The peak force in the SAS device by the equivalent linear model is almost half of the non-linear model. This is not an unexpected phenomenon, and the equivalent linear model will predict the peak force in the SAS in this way only as the equivalent stiffness of the SAS is half of the stiffness of the corresponding non-linear model (refer to equation (7)). The peak force in the SAS device is expected to come under the maximum displacement condition, in which the velocity is zero. Thus, for a maximum displacement, the peak force predicted by the equivalent linear model will be half of the non-linear model. Thus, the equivalent stiffness and damping approach of a base-isolated structure with the SAS for the earthquake time history analysis predicts the highest floor total acceleration, isolator displacement, the force in the isolation system, and the entire base shear with good accuracy. However, the predicted force in the SAS device is roughly half of the non-linear analysis.

Table 2.

Comparison of the response of base-isolated structures with the SAS using non-linear analysis with the corresponding equivalent stiffness and damping approach.

α	${\ddot{x}}_{N}^{a}$ (g)	x_b (m)	F_iso/W	F_sas/W	F/W	${\ddot{x}}_{N}^{a}$ (g)	x_b (m)	F_iso/W	F_sas/W	F/W
α	Non-linear analysis					Equivalent approach
Northridge, 1994
0.1	0.592	0.464	0.479	0.047	0.524	0.548	0.464	0.478	0.028	0.504
0.2	0.74	0.402	0.419	0.202	0.617	0.599	0.4	0.414	0.119	0.528
0.3	0.856	0.378	0.394	0.266	0.655	0.61	0.376	0.388	0.154	0.539
Imperial Valley, 1979
0.1	0.35	0.29	0.297	0.029	0.326	0.349	0.295	0.301	0.017	0.318
0.2	0.373	0.219	0.225	0.11	0.334	0.339	0.233	0.238	0.066	0.302
0.3	0.385	0.195	0.201	0.138	0.337	0.335	0.211	0.217	0.084	0.297
Kobe (Japan), 1995
0.1	0.439	0.357	0.369	0.036	0.404	0.406	0.357	0.369	0.022	0.389
0.2	0.579	0.299	0.312	0.151	0.46	0.439	0.303	0.312	0.09	0.398
0.3	0.662	0.274	0.286	0.193	0.476	0.443	0.276	0.285	0.113	0.395
Loma Prieta, 1989
0.1	0.617	0.494	0.511	0.05	0.559	0.597	0.498	0.515	0.031	0.544
0.2	0.753	0.382	0.397	0.192	0.583	0.576	0.398	0.413	0.123	0.53
0.3	0.784	0.341	0.356	0.24	0.59	0.563	0.361	0.376	0.154	0.525

Conclusions

The investigation into the behaviour of base-isolated structures with supplemental SAS (Seismic Absorption Systems) has yielded the following key findings:

1. SAS is a more effective supplemental device for base-isolated structures than the SS (Seismic Supplementary) device. It significantly reduces both floor accelerations and isolator displacements.

2. Under stochastic broad-band earthquake excitations, there is an optimal stiffness value for the supplemental SAS devices at which the RMS (Root Mean Square) absolute acceleration reaches its minimum.

3. This optimal stiffness value for the SAS decreases as the isolation period and damping of the isolation system increase.

4. The proposed equations for predicting the RMS isolator displacement, top floor absolute acceleration, and optimal stiffness of the SAS accurately estimate the expected response. These equations can be effectively used in the preliminary design of base-isolated structures with SAS devices.

5. The pattern of isolator displacement in base-isolated structures supplemented with SAS devices under actual near-fault earthquake records shows a high correlation with the outcomes from stochastic earthquake excitation analyses.

6. The equivalent stiffness and damping approach for base-isolated structures with SAS devices used in earthquake time history analysis accurately predicts the maximum floor accelerations, isolator displacements, forces within the isolation system, and total base shear. However, it should be noted that the predicted force within the SAS device is approximately half of that obtained from a non-linear analysis.

This study has demonstrated the potential of SAS devices to enhance the seismic performance of base-isolated structures by optimizing key parameters such as stiffness, which can be critical for minimizing accelerations and displacements during seismic events. Additionally, the SAS outperforms a simple spring in mitigating the dynamic response of isolated structures. However, these findings need to be validated through experimental testing. Furthermore, the performance of the SAS should be evaluated considering the non-linear behaviour of isolated structures and compared with classical dampers such as viscous and friction dampers. The effectiveness of the SAS in reducing the seismic response should also be examined, taking into account the effects of soil-structure interaction.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

R.S. Jangid

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Seismic analysis and performance of semi-active spring for base-isolated structures

Abstract

Keywords

Introduction

Base-isolated structure with SAS

Mathematical modelling of SAS

Governing equations of motion

Response under filtered white-noise earthquake excitation

Approximate response and optimum stiffness of SAS

Flexible isolated structure under near-fault earthquakes

Comparison of the linear and non-linear response

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References