A combined control strategy using tuned liquid dampers to reduce displacement demands of base-isolated structures: a probabilistic approach

Abstract

This paper investigates a hybrid structural control system using tuned liquid dampers (TLDs) and lead-rubber bearing (LRB) systems for mitigating earthquake-induced vibrations. Furthermore, a new approach for taking into account the uncertainties associated with the steel shear buildings is proposed. In the proposed approach, the probabilistic distributions of the stiffness and yield properties of stories of a set of reference steel moment frame structures are derived through Monte-Carlo sampling. The approach is applied to steel shear buildings isolated with LRB systems. The base isolation systems are designed for different target base displacements by minimizing a relative performance index using Genetic Algorithm. Thereafter, the base-isolated structures are equipped with TLDs and a combination of the base and TLD properties is sought by which the maximum reduction occurs in the base displacement without compromising the performance of the system. In addition, the effects of TLD properties on the performance of the system are studied through a parametric study. Based on the analyses results, the base displacement can be reduced 23% by average, however, the maximum reduction can go beyond 30%.

Keywords

tuned liquid damper / lead-rubber bearing system / probabilistic framework / steel shear building / relative performance index / Monte-Carlo sampling

Introduction

In the last decades, various structural control systems are successfully employed to enhance the performance of structures under wind or seismic excitations. These control systems regulate the dynamic response of structures through supplying additional damping, energy dissipation, and adding to the inertia force of the system. Many studies investigated different control strategies such as braced frames [¹], viscous dampers [²], self-centering systems [³], and tuned mass dampers (TMD) [⁴] to reduce the displacement demands of structures. Among passive control systems, seismic base isolation is one of the most effective techniques to reduce earthquake-induced forces and to enhance the seismic performance of structures [⁵–⁷]. Various isolation systems are developed including elastomeric-based bearings [⁸,⁹] and isolation systems based on sliding, e.g., friction pendulum systems [¹⁰]. A great deal of research has been dedicated to the implementation of different isolation devices and examining their performance under different loading conditions. Regarding isolation systems based on sliding, Castaldo and Tubaldi [¹¹] studied the seismic performance of a wide range of base-isolated structures equipped with friction pendulum systems (FPS). Furthermore, the effects of different properties of the friction pendulum bearings on the reliability of isolated structures were studied by [¹²–¹⁵].

Elastomeric-based isolation systems are widely used to protect structures against earthquake or wind-induced vibrations. These systems mitigate the vibration of the super-structure through providing additional damping, dissipating energy through hysteresis, and concentrating large deformations onto the bearings. Several studies aimed at addressing different aspects of the behavior of such systems. Tubaldi et al. [¹⁶] studied the softening behavior of high damping natural rubber (HDNR) bearings through experiment. Tubaldi et al. [¹⁷] investigated the effects of uplift in rubber steel laminated bearings when subjected to tensile loads in multi-span isolated bridges. Cancellara and De Angelis evaluated the performance of RC structures isolated with lead-rubber and high damping rubber bearings [¹⁸–²⁰].

Although base isolation systems are effective in reducing structural and non-structural damage to the super-structure, the limitations on the base displacements of these isolation systems, however, limit the amount of performance improvement that can be practically obtained. In fact, this is among the most important design parameters of base isolation systems. To reduce the displacement demands at the flexible isolation layer, it has been suggested to implement dynamic vibration absorbers.

TMD and tuned liquid dampers (TLDs) are dynamic vibration absorbers that have been employed extensively in order to suppress the vibrations of structures. Several researchers have investigated the implementation of TMDs or Active TMDs to reduce the displacement demands of different structures including base-isolated structures subjected to wind or seismic excitations [⁴,²¹,²²]. Tuned liquid damper is an effective passive control device that dissipates energy through wave breaking and the sloshing motion of the liquid. This damper has been investigated extensively and a great piece of research has focused on capturing the dynamic behavior of the liquid inside the TLD tank by proposing nonlinear sloshing models [²³–²⁷], equivalent TMD models [²⁸–³⁰], models based on finite element method [³¹], and novel techniques such as real-time hybrid simulation (RTHS) [³²,³³].

The effects of TLDs on the response of structures were investigated in several studies. Banerji et al. [³⁴] studied the feasibility of utilizing rectangular TLDs to mitigate the vibrations of structures with various natural periods and damping ratios. Tait et al. [³⁵] studied the effects of different parameters on the performance of structure-TLD system subjected to wind excitation. Ashasi-Sorkhabi et al. [³⁶] experimentally investigated the effects of mass ratio, the ratio of TLD mass to the structure total mass, and frequency ratio, the ratio of TLD frequency to the frequency of the structure, on the TLD-structure system in resonance condition under harmonic excitation. Samanta and Banerji [³⁷] conducted a numerical study on employing TLDs in single- and multi-degree-of-freedom systems. Love and Tait [³⁸] investigated the implementation of multiple tuned liquid dampers (MTLDs) to enhance the robustness of the system. Wang et al. [³⁹] studied the TLD-structure systems using real-time hybrid simulation (RTHS) technique. Love et al. [⁴⁰] proposed a hybrid structural control system using TLDs to reduce wind-induced motion of a base-isolated structure in which the TLD was modeled by an equivalent mechanical model [²⁹,⁴¹] and the base-isolated structure was represented by a linearized Bouc-Wen model.

It is well established that uncertainties have a substantial influence on different aspects of an engineering problem. In this regard, many studies aimed at taking into account the stochastic nature and the uncertainties associated with different engineering problems [⁴²–⁴⁶]. In the field of structural control, the optimum properties of the system are influenced by the uncertainties in the variables of the problem. In fact, neglecting the uncertainties may lead to an unsafe design. Therefore, many researchers have put forth effort to design control systems through a probabilistic approach. Son and Savage [⁴⁷] presented a non-sampled-based probabilistic approach for the design of vibration absorbers using first-order reliability method (FORM), which was validated by Monte-Carlo sampling. Yu et al. [⁴⁸] investigated the reliability-based design of TMDs by using a reliability sensitivity-based approach. Debbarma et al. [⁴⁹] investigated the optimization of tuned liquid column damper parameters considering uncertain but bounded (UBB) system parameters. Scruggs et al. [⁵⁰] proposed a probability-based design approach for active base isolation systems.

This paper aims at evaluating the performance of inelastic base-isolated structures equipped with TLDs under seismic excitation. Furthermore, a probabilistic approach for considering the uncertainties in steel shear buildings is proposed. In the proposed approach, the uncertainties related to the inelastic steel shear building superstructure are taken into account by using the probability density function (PDF) of the stiffness and the yield properties of the stories and also the correlations between the aforementioned properties. A set of steel moment-resisting frame structures with different number of stories are selected. The probabilistic model of the reference structures are established in OpenSees [⁵¹] in accordance with the Joint Committee on Structural Safety (JCSS) probabilistic code [⁵²]. Thereafter, each structure is subjected to a large number of pushover analyses and the distributions of the stiffness and yield properties of stories are derived. The proposed approach is applied to steel shear building structures isolated with lead-rubber bearing (LRB) systems. The base isolation systems are optimally designed by minimizing a relative performance index (RPI) for a given target base displacement using Genetic Algorithm. To characterize the liquid motion inside the TLD tank, a finite element model [³¹] is employed. Then, the base-isolated structures are equipped with TLDs to minimize the base displacement under a set of natural ground motion records without compromising the performance of the system. In the final part, a parametric study is carried out to investigate the effects of TLD properties on the performance of the reference base-isolated structures.

Structural models

In this study, three steel moment-resisting frame structures located in Los Angeles with three, six, and nine stories are selected from the SAC steel project [⁵³] and a study by [⁵⁴]. The structural system for these buildings consists of steel perimeter moment-resisting frames with interior gravity frames. In addition, all structures are office buildings and they are designed to conform to the local code requirements. The structures are modeled as two-dimensional frames using OpenSees [⁵¹] and half of the seismic mass at each floor is assigned to the frame. Furthermore, a basic center line model of the structure is developed and the plasticity of the members is modeled by fiber elements. Additionally, the influence of the interior gravity frames are taken into account through introducing P-D columns attached to the frame by rigid elements. The modal properties of the structures are provided in Table 1.

Proposed probabilistic approach

The paper puts forward an approach to take into account the uncertainties associated with shear building model properties including story stiffness and post-yield parameters. For this purpose, the probabilistic model of the reference structures are established in OpenSees according to JCSS [⁵²]. The sources of randomness, i.e., the uncertainties in section dimensions, material yield properties, and loading conditions are presented in Table 2. In Table 2, A_nominal denotes the nominal value of the member cross section, E_sp is the nominal value of steel modulus of elasticity, and f_y refers to the steel yield stress calculated as below:

(1)f_y=f_{y_{sp}}\cdot \alpha \cdot e^{(-u\cdot \nu )}-\mathrm{C},

where f_ysp refers to the nominal value for the steel yield stress, a is spatial position factor (a = 1.05 for the web of hot rolled sections and a = 1 otherwise per [⁵²]), u is a factor ranging between -1.5 and -2 that takes into the difference between the nominal value and the mean value, n denotes the Poisson’s ratio, and C is a constant equal to 20 MPa per [⁵²]. Furthermore, the live load, q, is modeled as a random variable with the mean of m_q = 0.5 kPa and the standard deviation of 0.3 kPa per [⁵²].

Different sampling techniques such as Latin Hypercube Sampling (LHS), Quasi-random sequences, and Monte-Carlo Sampling are employed for uncertainty quantification in several studies [⁵⁵–⁵⁸]. This study uses Monte-Carlo Sampling and a total number of 4000 samples are generated from the PDF of the properties presented in Table 2 and a nonlinear static pushover analysis is performed on each sample. In the pushover analysis, each story is pushed until it reaches the drift ratio of 2.5%, which is the life-safety drift ratio limit for steel moment-resisting frames [⁵⁹]. The resulting capacity curve is replaced by an idealized bilinear curve in accordance with [⁶⁰] and the stiffness and the yield properties of stories including yield shear and post-yield stiffness ratio are derived. Thereafter, a distribution which best fits the value of the samples is sought. For this purpose, goodness of fit tests are conducted on each of the candidate distributions with the confidence interval defined as follows [⁶¹]:

(2){\langle \mu \rangle }_{\mathrm{1}-\alpha }=\left[\overline{x}-t_{\alpha /\mathrm{2},n-\mathrm{1}}{\displaystyle \frac{s}{\sqrt{n}}};\overline{x}+t_{\alpha /\mathrm{2},n-\mathrm{1}}{\displaystyle \frac{s}{\sqrt{n}}}\right],

where a is the confidence level, \bar{x} is the samples mean, t_{\alpha /\mathrm{2},n-\mathrm{1}}refers to the student’s t-distribution value with (n− 1) degrees of freedom, evaluated at the probability of (1 −a/2), and s denotes the samples standard deviation. In this study, the confidence level of 0.99 is selected and the number of samples is determined such that the maximum difference between the samples mean and the upper or lower confidence limit lies beneath 1%.

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Story stiffness

As it was discussed in the previous section, a total number of 4000 Monte-Carlo samples are subjected to pushover analysis and the stiffness of stories are determined from the idealized capacity curve. Then, a distribution that best describes the stiffness of stories is sought. After comparison between different distributions, it is found that a lognormal distribution best fits the stiffness values. Figures 1(a)–1(c) compare the empirical CDF and the fitted lognormal CDF values for each story of the 3-story building model. The results for other structures are not presented due to space constraints.

Table 3 shows the mean value and the coefficient of variation (CoV) of the stiffness of stories. According to Table 3, to account for the uncertainties in the stiffness of stories, a lognormal random variable should be defined with its mean equal to the story stiffness and a coefficient of variation of 3%. Figure 2 shows the correlation coefficients, r, between the stiffness of stories of the reference models. As seen, the correlation between the stiffness of consecutive stories decreases with moving along the height of the structure. The average correlation coefficient between the stiffness of stories for the 3-story, 6-story, and 9-story models are 0.78, 0.73, and 0.65, respectively. Based on the results, an average correlation coefficient of 0.74 can be assumed to take into account the correlation between stiffness of stories.

Yield properties of stories

In this section, the distributions for the yield properties of stories including yield shear, V_y, and post-yield stiffness ratio, a, are derived. The properties of the distributions for V_y and a are presented in Tables 4 and 5, respectively.

Moreover, the statistical analysis demonstrated that a lognormal distribution best describes both V_y and a. Figure 3 demonstrates the correlation matrix for V_y. The same pattern for the correlation coefficients between the yield shear and post-yield stiffness ratio of stories is observed. The average correlation coefficient between the yield shear of stories for 3-story, 6-story, and 9-story models are 0.86, 0.72, and 0.64 (average=0.74), respectively. The corresponding values for the post-yield stiffness ratio of stories are 0.83, 0.78, and 0.74 (average= 0.78), respectively. Moreover, according to Fig. 4, the average correlation coefficient between Vy and a of different stories, i.e., r_ij is approximately 50%, while this value for V_y and a of the same story, i.e., r_ij, is about 70%. Eventually, based on the analyses results, it is concluded that in order to take into account the uncertainties within the yield properties of stories, one should assign a lognormal distribution to both V_y and a, with its mean equal to the value of the yield shear and post-yield stiffness ratio of stories obtained from the pushover analysis and a coefficient of variation of 0.06 and 0.15 for V_y and a, respectively. The distributions derived in this section, facilitate establishing the probabilistic framework for the analysis of nonlinear shear building structures, which are common models of structures in structural dynamics.

Analysis and Results

The equation of motion of a multi-degree-of-freedom (MDOF) system (Fig. 5) is given by:

(3)\mathit{M}\ddot{\mathit{u}}+\mathit{C}\dot{\mathit{u}}+\mathit{R}=-\mathit{M}\mathit{\iota }{\ddot{u}}_g

In Eq. (3), M denotes the mass matrix and C is the Rayleigh damping matrix. It should be noted that a damping ratio of 0.02 is assumed for the structure and the damping matrix is constructed by setting the damping ratio of the first mode and the mode with the period of 0.2 s equal to 0.02. The vector of displacements is represented by u, ι is the influence vector, and {\ddot{u}}_grefers to the ground acceleration. Moreover, R is the vector of restoring forces of stories and its ith component is calculated according to (4) [⁶²]:

(4)\begin{array}{l}{R}_i={\alpha }_i{\displaystyle \frac{f_{y_i}}{u_{y_i}}}{u}_i(t)+(1-{\alpha }_i)f_{y_i}{z}_i(t),\\ {\dot{z}}_i(t)={\displaystyle \frac{1}{u_{y_i}}}\left[{\dot{u}}_i(t)-\mathit{\gamma }\left|{\dot{u}}_i(t)\right|z_i(t){\left|z_i(t)\right|}^{n-\mathrm{1}}-\beta {\dot{u}}_i(t){\left|z_i(t)\right|}^n\right]\end{array}

where, R_i is the restoring force of the ith story. The post-yield stiffness ratio, yield shear, and yield displacement of the ith story are represented by a_i, f_yi, and u_yi, respectively. The parameter that accounts for the hysteresis behavior is represented by z and g, b, and n are the Bouc-Wen model parameters, which determine the shape of the hysteresis loops. Figure 5 shows a schematic view of the MDOF system. The mass and the stiffness of the ith story are represented by m_i and k_i, respectively.

After establishing the probabilistic model of the structure based on the results obtained in section ‎3, each structure is subjected to 20 natural ground motion records representative of a 10% probability of exceedance in 50 years for Los Angeles [⁶³], scaled to the ASCE 7-10 [⁶⁴] design spectrum for stiff soil. Equations. (3)–(4) are solved simultaneously in order to find maximum drift ratios and absolute accelerations of stories. Subsequently, a distribution that best fits the aforementioned parameters is derived. It was found that a lognormal distribution best describes the aforementioned quantities. The properties of the distributions are provided in Table 6, where r denotes the correlation coefficient between the maximum drift ratio and the maximum absolute acceleration. The obtained correlation coefficient indicates that when the maximum drift ratio increases due to higher excitation intensities, the value of the maximum absolute acceleration increases as well.

Design of base-isolated structures with LRB systems

This section briefly summarizes the design procedure of the LRB systems designed for the target base displacements of 200, 250, and 300 mm for each building model. The base isolation system is modeled as an equivalent bilinear spring, whose properties including the total initial stiffness, k_1t, the total post-yield stiffness, k_2t, and the total yield force, F_yt, are determined such that the best performance of the system is achieved for a given target base displacement. A comprehensive definition on the design of base-isolated structures can be found in Ref. [⁶⁵]. After performing a preliminary design and obtaining the values of k_1t and F_yt, an appropriate range for each property is assumed, e.g., 50% to 150% of the initial values. Then, the optimal values of the base isolation design variables are determined through an optimization process. Gradient-based methods [⁶⁶] and meta-heuristic algorithms such as Genetic Algorithm [⁶⁷] are among the optimization methods that are extensively implemented in optimization problems. In this study, Genetic Algorithm is employed to find the optimum set of isolation properties. Genetic Algorithm requires an objective function, which is evaluated for randomly generated design variable values. Those candidate values that yield the lowest function value are selected as the basis of the next generation. This process continues until the variations of the objective function satisfies a pre-defined function tolerance. In this paper, the objective function is defined as below:

(5)f(k_{\mathrm{1}t},F_{yt})=\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left({\displaystyle \frac{a_{\mathrm{B}{\mathrm{I}}_{\max }}(k_{\mathrm{1}t},F_{yt})}{a_{{\mathrm{0}}_{\max }}(k_{\mathrm{1}t},F_{yt})}}+{\displaystyle \frac{{\delta }_{\mathrm{B}{\mathrm{I}}_{\max }}(k_{\mathrm{1}t},F_{yt})}{{\delta }_{{\mathrm{0}}_{\max }}(k_{\mathrm{1}t},F_{yt})}}\right)\\ \left|{\displaystyle \frac{x_b(k_{\mathrm{1}t},F_{yt})-x_{b_{\mathrm{target}}}}{x_{b_{\mathrm{target}}}}}\right|\end{array}\right\}\mathrm{.}

The first term in Eq. (5) is referred to as the RPI [⁶⁸] in which a_BImax and d_BImax are the mean value of the maximum absolute acceleration and the maximum drift ratio of the base-isolated structure, respectively. a_0max and d_0max represent the average maximum absolute acceleration and the average maximum drift ratio of the uncontrolled structure, respectively. The second term deals with the variations of the maximum base displacement of the isolation level (x_b) and guarantees that the system is designed for the target base displacement (x_btarget). In this study, a function tolerance equal to 0.0001 is selected to make sure that the optimization is carried out with high accuracy. The results of the optimization and the optimum design variables are presented in Fig. 6 and Table 7, respectively.

Tuned liquid damper (TLD)

The model employed for characterizing the liquid motion is the Simplified Sloshing Model (SSM), which was introduced and validated by Ruiz et al. [³¹]. This model is computationally efficient and enjoys high accuracy as well, facilitating its implementation in parametric studies. Figure 7 shows a 2D scheme of the tuned liquid damper with the tank length L, water initial depth H. The volume of the fluid is represented by W. Furthermore, G_s denotes the free surface at any time t, G_p is the surface of walls and bottom, and G₀ represents the non-perturbed free surface (z = 0). In addition, an auxiliary coordinate h is defined, which measures the relative displacement between the free surface and the coordinate system.

In this model, the continuity and equilibrium equations govern the motion of the fluid and by defining a velocity potential function, j, they take the form of Laplace and Bernoulli equations. By using finite element method, the sloshing problem is interpreted as an equivalent mass-spring model in which damping is taken into account by introducing a damping matrix. The modal damping ratio is selected equal to 0.5%, which is a common damping ratio for TLDs [³¹]. The nodal forces of the ith element is determined by (6):

(6){{\mathit{F}}_{\mathit{p}}}^{(i)}-\sin \left({\alpha }^{(i)}\right)\underset{{{\mathit{\Gamma }}_p}^{(i)}}{\int }{{{\mathit{N}}_{\mathit{\eta }}}^{(i)}}^T{{\mathit{N}}_{\mathit{\eta }}}^{(i)}\mathrm{d}{{\mathrm{\Gamma }}_{\mathrm{p}}}^{(i)}{{\mathit{P}}_{\mathit{p}}}^{(i)},

where, G_p⁽ⁱ⁾ is the surface of the ith element that coincides with the tank walls and the bottom. a⁽ⁱ⁾ is the angle between G_p⁽ⁱ⁾ and x-y plane. In addition, N_η is the weighting function of the nodes located on the free surface and P_p is the vector of nodal pressure. Subsequently, the total transmitted force, F, is calculated by summing the nodal forces.

Parametric study

To find the optimal properties of the hybrid control system and also to investigate the effects of TLD parameters, a parametric study is performed. In the parametric study, the isolation design variables, i.e., k_1t and F_yt, are varied in the range of 50% to 200% of their corresponding optimal values. Furthermore, the mass ratio m, of the TLD is varied between 0.01 and 0.05 and the frequency ratio b, which is the ratio of the TLD frequency to the frequency of the base-isolated structure is varied between 0.5 and 1.5. Each combination of the aforementioned properties is subjected to the selected ground motion records and the base displacement and the RPI of the system are evaluated. The equation of motion of the combined (Fig. 8) reads:

(7)\mathit{M}\ddot{\mathit{u}}+\mathit{C}\dot{\mathit{u}}+\mathit{R}=-\mathit{M}\iota {\ddot{\mathbf{u}}}_g+{\mathit{L}}_{\mathrm{s}}F

where, L_s is the vector of the TLD location, which is a vector of zeroes except the component corresponding to the TLD floor being equal to unity. The hybrid control system is designed such that the maximum reduction in the average maximum base displacement occurs, while the difference between the average RPI of the system and the RPI of the corresponding optimally-designed base-isolated structure, stays below a pre-specified bound, which is selected 10% in this study. The analysis results are presented in Figs. 9–10 and the optimum properties are provided in Table 8.

Figure 9 demonstrates the effects of mass ratio on the RPI of the system for a constant frequency ratio. It is observed that the effect of mass ratio varies for different sets of system properties. However, in most cases the higher mass ratio results in a higher RPI, which is not desirable. For example, in the case of the 9-story building model, the negative effects of higher mass ratios are more pronounced. On the other hand, for the 6-story model these effects vary widely such that for some combinations of the system properties, lower m leads to a better performance and in some cases the opposite is true. Nevertheless, based on the results, lower m is more desirable, which is in agreement with other studies [³⁶] and practice suggesting that the mass ratio should be kept between 1% and 3%. Figure 10 shows the effects of different frequency ratios on the performance of the system. It is observed that as the natural frequency of the TLD deviates from the natural frequency of the structure, the effectiveness of the TLD tends to decrease. As a case in point, for the 3-story building model as the frequency ratio decreases and it becomes closer to 1, the performance of the system improves.

Furthermore, it is observed that TLD shows the highest efficiency in the case of 9-story building. In this model due to the larger seismic weight and more contribution of the higher modes, TLD is more excited and it generates larger forces. As a result, reduction is more pronounced compared to the other building models; however, the effectiveness of the base isolation is not significant in this case.

Table 8 shows that as the isolation level becomes stiffer and it yields at higher values of F_yt, the effectiveness of TLD increases. For the sake of illustration, in the case of the 6-story building, the highest reduction has occurred for x_btarget = 200 mm and as it seen it has the highest yield force compared to x_btarget = 250 mm and x_btarget = 300 mm. This behavior is due to the fact that larger forces are produced in stiffer systems and as the isolation level becomes stiffer and less ductile, larger forces are imposed on the TLD, hence, TLD is more excited and its performance escalates.

Conclusions

In this study, a combined control strategy using TLDs and base isolation systems under earthquake excitation was investigated. Furthermore, a probabilistic approach was proposed for probabilistic modeling of steel shear building structures. In the proposed approach, the distributions of the stiffness and yield properties of stories were derived through a Monte-Carlo sampling procedure. The proposed approach was applied to three steel moment frame structures with different number of stories. Moreover, the uncertainties associated with the seismic input were taken into account by employing a set of natural ground motion records, compatible with the characteristics of the reference site. Base-isolated structures with target base displacements of 200, 250, and 300 mm were optimally designed using Genetic Algorithm. Thereafter, the base-isolated structures were equipped with TLDs and a parametric study was carried out to investigate the effects of TLDs on the performance of base-isolated structures. In the parametric study, a combination of TLD and isolation properties was sought by which the maximum reduction occurred in the base displacement, while the variations of the performance of the system, i.e., RPI was kept below 10%. Furthermore, the effects of mass ratio and frequency ratio were investigated. It was concluded that TLD is more effective for mass ratios between 1% and 3%. Regarding the effects of the frequency ratio, as the frequency ratio deviated from unity, the performance of TLD was reduced. According to the analysis results, b = 1.3 for 3-story and 9-story buildings and b = 0.7 for 6-story yielded the best performance. In conclusion, the proposed control strategy has shown to be effective in reducing the displacement demands at the isolation level. It was shown that the base displacement can be reduced 23% by average, but the maximum reduction can go beyond 30%.

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