Fuzzy sliding mode control of nonlinear smart base isolated building under earthquake excitation

Transcription

1 Received: 18 February 2018 Revised: 13 July 2018 Accepted: 2 September 2018 DOI: /tal.1557 RESEARCH ARTICLE Fuzzy sliding mode control of nonlinear smart base isolated building under earthquake excitation Hosein Ghaffarzadeh 1 Amir Hossein Ghaffari 1 T. Y. Yang 2 1 Department of Civil Engineering, University of Tabriz, Tabriz, Iran 2 Department of Civil Engineering, University of British Columbia, Vancouver, Canada Correspondence Hossein Ghaffarzadeh, Department of Civil Engineering, University of Tabriz, Tabriz, Iran. ghaffar@tabrizu.ac.ir Summary In this paper, the effectiveness of the fuzzy sliding mode control strategy on three dimensional benchmark building with smart base isolation under seismic excitation has been examined. One of the appropriate control theories for such this nonlinear system is the sliding mode control theory; discontinuous sliding mode theory has weakness such as chattering phenomena. In this paper, we used a combination of fuzzy logic and sliding mode theory for the deletion of this defect. The proposed control theory has been scrutinized by applying on lately developed nonlinear three dimensional base isolated benchmark building. This building because of the three dimensional nature, coalescing of lateral and torsional responses, continuity of responses of the superstructure, and base is modeled with three degrees of freedom on every floor. In this building eight actuators assigned only at the base level and in the two directions (x, y). In other words, 16 actuators are located underneath the structure. Furthermore, the base isolation system has been modeled by considering lateral coupled equations for a better examination of the performance of the system. The results indicate that reduction of control performance is remarkable. Also, utilizing proposed control theory can decrease the responses of building in two main directions and, particularly, in the rotational degree of freedom. KEYWORDS 3D building modeling, Bouc Wen model, earthquake excitation, fuzzy logic, sliding mode, smart base isolation 1 INTRODUCTION Structural control has become an attractive field for preserving the buildings from environmental vibrations in recent years. The main scope of structural control is to mitigate wind and earthquake induced forces on the entire structure by changing dynamic structural properties such as mass, stiffness, or damping. Control systems consist of passive, active, semiactive, or their combination. Base isolation is one of the efficient ways to protecting buildings under the earthquakes that is very used for decreasing the structural responses. The base isolation consists mostly of friction, lead rubber bearing, and linear bearing. The main idea of the function of base isolation is based on increasing the structure's flexibility for avoiding seismic ground motion. [1 5] Passive base isolated systems in some earthquakes may cause large base displacements according to increased flexibility and cannot reduce displacements of building perfectly. [6,7] For this reason, adding the other control utensil, in order to the elimination of this disadvantage, is required. Furthermore, Smart base isolation can amend the performance of not only superstructure but also base level. Under certain circumstances, the smart base isolation that contains passive base isolation and another control system for instance, active control system was employed. [8 12] The smart base isolation system provides improved performance compared with conventional passive base isolation systems. During the recent years, many researchers have worked on this combination. Narasimhan et al. [13] presented smart base isolation system for seismically excited benchmark building. They modeled three types of base isolators by using the Bouc Wen model. Suresh Struct Design Tall Spec Build. 2019;28:e wileyonlinelibrary.com/journal/tal 2018 John Wiley & Sons, Ltd. 1of14

2 2of14 et al. [14] demonstrated adaptive control theory based on Gaussian kernel function for nonlinear base isolated buildings. In this study, the stability of scheme is evaluated with the Lyapunov approach. The proposed method did not require to estimate the system parameters and to carry out any iteration for acquiring the control force. Narasimhan et al. [15] employed nonlinear base isolation system and eight controllers at the base level to decrease the dynamic responses. A semiactive algorithm for reduction of seismic responses of buildings using a variable orifice dampers is proposed by Ghaffarzadeh et al. [16] The results of the proposed method indicated the proper performance of the strategy. Kim and Roschke [17] utilized the fuzzy logic controller in a three dimensional building with smart base isolation to mitigate dynamic responses of a structure under excitation of seven earthquakes. Furthermore, in this study, the genetic algorithm was employed to optimize the fuzzy control technique. Gu et al. [18] aimed the performance of new frequency control theory with magneto rheological elastomer base isolator in a five story building. The proposed algorithm extended for averting the resonant state. Accordingly, the results of the simulation showed the significant diminution in structural responses. Alli and Yakut [19] presented a robust fuzzy sliding mode controller for a structure with base isolation and indicated that this method is a useful theory in comparison of conventional SMC. Li et al. [20] studied nonlinear structural vibration with the adaptive fuzzy sliding mode control algorithm. Chang and Spencer [21] investigated the development and experimental verification of an active base isolation system under the earthquake excitation. A new control algorithm with variable friction semiactive was developed to enhance the performance of the smart base isolated buildings by Narasimhan and Nagajaiah. [22] The H controller was applied to ascertain the proper control forces and to avoid the rapid switching of the variable friction devices. Van Engelen et al. [23] examined modifying the stiffening regime of Stable unbonded fiber reinforced elastomeric isolators by using modified support geometry. The results of this study show that modified support geometry can maintain the stability and provide control over the stiffening of stable unbonded fiber reinforced elastomeric isolators. Habieb et al. [24] scrutinized the behavior of the unbonded fiber reinforced elastomeric isolators under large displacements by 3D finite element model. Furthermore, in this paper, different kinds of isolators such as bonded, unbonded, and friction between pads have been compared. This study demonstrates that unbonded case has the best performance among selected isolators. Subsequently, this paper presents fuzzy sliding mode on three dimensional benchmark building with the smart base isolation that contains passive base isolation and active control system for seismically excitation. Sliding mode control theory is extended for exerting to the threedimensional building, and nonlinear system and fuzzy logic compounded the sliding mode control theory for approaching the reduction of the unfavorable property of sliding mode control theory. The torsion in the building is one of the elements that exacerbated the response of building due to the cohesiveness of degrees of freedom (DOFs). For this reason, the benchmark building is modeled three DOFs per floor at the center of mass, and smart base isolation system is assumed nonlinear. Based on the latter premise, Bouc Wen model was used for simulation of the behavior of nonlinear smart base isolation system. In this building, eight actuators allocated only at the base level and in the two main directions (x, y). We intended to diminish of main DOF response with using the actuators for reduction besides that, the reduction of response of rotational DOF is considered. Finally, the proposed strategies are shown improvement in dynamic responses of an extensive range of seismic excitation. 2 CONFIGURATION OF STRUCTURAL MODEL The structural model investigated here is similar to an existing building in Los Angeles, California. [13] The benchmark building is a base isolated eight story, steel braced frame building, L shaped in the plan, 82.4 m length, and 54.3 m wide as shown in Figure 1. In this structure, the system of base isolators is connected to superstructure by the rigid concrete slab, as shown in Figure 1. The superstructure is modeled as three dimensionally and is assumed to remain elastic. Floor and base slabs are considered to be rigid on the plane. All levels of building consist of superstructure and base are designed by assuming 3 DOFs per floor at the center of mass. Then, 27 DOFs consist of 24 DOFs for the superstructure, and 3 DOFs for isolation system are considered in modeling. All 24 modes of vibrations in the fixed base building are utilized in the modeling of the superstructure. It is to be noted that the damping ratio of the superstructure is considered 5% for all fixed base mode. 3 MODEL OF BASE ISOLATION SYSTEM There are different kinds of the base isolation system. In practice, a combination of two or more kinds of base isolation can be used to model base isolation system. In this study, the isolation system included a combination of elastomeric bearings as elastic elements and friction pendulum bearings as hysteretic elements. Indeed, the smart base isolation system consists of 92 base isolation, which 61 of them are friction pendulum bearings and others are elastomeric bearings. The elastomeric bearing is one of the isolators with low damping. The damping of these kinds of isolators varies around 3%. Also, stability against temperature and manufacturing cost are the two main advantages of elastomeric isolators. On the other hand, the friction isolators are influenced by vertical loads and friction forces. This mechanism leads to lower friction coefficient, whereas a higher vertical load is applied. Besides, one of the main disadvantages of these types of isolators is that friction isolator can have a proper performance for a specific level of ground motion. In other words, friction base isolation has a long period and may come in resonance. Based on these reasons, the proposed combination attempts to enhance the performance of base isolation system. There are numerous methods to model base isolation system. Actually, the behavior of isolators can be assumed uniaxial or biaxial. In practice, base isolation system experiences multidirectional motion under the earthquake excitation. Furthermore, according to the analytical studies, the biaxial interaction significantly

3 3of14 FIGURE 1 (a) Isolation plan; (b) Finite element model of superstructure; (c) elevation of devices contains actuator and isolation (Narasimhan et al. [13] ) affects the response in the nonlinear range. Therefore, the Bouc Wen biaxial model was used to capture the accurate behavior of the base isolation system. [25 28] _z x ¼ A_u x β _u x z x jz x γ _u x z 2 x β j _u yz y jz x γ _u y z x z y ; _z y ¼ A_u y β _u y z y jz y γ _u y z 2 y β j _u xz x jz y γ _u x z x z y ; (1) where Z x and Z y are dimensionless hysteretic variables that are bounded by values ±1. A, β, and γ are dimensionless quantities; A/(β + γ) =1;u x, u y and _u x ; _u y represent the displacements and velocities, respectively, in the x and y directions at the base level; and u y is the yield displacement. The force in isolation system could obtain from below equations [13] : f x ¼ k p U x þ c v Ux _ þ ðk e k p ÞU y z x ; f y ¼ k p U y þ c v Uy _ þ ðk e k p ÞU y z y ; (2) in which k e is the preyield stiffness, k p the postyield stiffness, and c v the viscous damping coefficient of the elastomeric bearing or device. The usage data for isolation system in this study are k e /k p = 0.142, k p = 919KN/m, u y = m, and c v = Also, the three dimensionless quantities A, B, γ are 1, 0.9, and 0.1, respectively. For friction bearings with flat or spherical sliding surface, Equation (1) was used, and the friction bearing force could obtain from Equation (3). f x ¼ k p U x þ μnz x ; f y ¼ k p U y þ μnz y : (3) In this building, dynamic responses under earthquake excitation can be exacerbated due to the L shaped plane of the structure. Due to analyzing the effectiveness and improving the performance of the structure under the earthquake excitation, 16 actuators at base level are placed. It should be noted that the capacity of each actuator is assumed 3,000 N. These actuators have been located in eight points according to the shape of the plane of the benchmark building, maximum control force of actuators and reducing the effect of the torsion in the benchmark building.

4 4of14 4 THREE DIMENSIONAL MODELING OF THE BENCHMARK STRUCTURE In this section, the benchmark structure is modeled as three DOF system. Due to the model linear superstructure, the equation of motion is extended by using fixed base properties. Furthermore, underneath of structure, each isolator is modeled with biaxial Bouc Wen model. Subsequently, the equation of motion for the superstructure is given by [13] M n n Un 1 þ C n n Un 1 _ þ K n n U n 1 ¼ M n n R n 3 Ug þ U b Þ 3 1 ; (4) in which n is 3 times the number of floors (excluding base) and M, C, and K denote mass, damping, and stiffness matrices of the superstructure, respectively. R is the matrix of earthquake influence coefficients. u; _u, and u represent the floor acceleration, velocity, and displacement vectors relative to the base, respectively. u b is the vector of base acceleration relative to the ground, and u g is the vector of ground acceleration. Besides, the equation of motion of the base level is expressed as follows: R T 3 n M n n h U n 1 þ R n 3 Ug þ U b i 3 1 n 1 þ M b3 3 Ug þ U b þ C _U b3 3 b3 3 þ K b3 3 U b3 3 þ f b3 1 þ f c3 1 ¼ 0; (5) 3 1 where, M b, C b, and K b are the diagonal mass matrix, resultant damping matrix of viscous isolation elements and resultant stiffness matrix of elastic isolation elements of the rigid base, respectively. Also, f b is control force that produced by bearings, and f c is control force of controllers. The controllers are located at the base level in eight locations, and each location contains two controllers in two directions. The placement of the controllers is shown in Figure 2. n By considering the combination of superstructure and base equations of motions and utilizing the state vector, X ¼ U T U T _ b U T U _ T, the equation of motion of combined system can be transformed to state space equation as following. [13] b g _XðÞ¼AX t ðþþbu t ðþþe t U g ðþ¼g t X; u; U g Þ; (6) where A, B, and E are condensed system matrices having 54 states derived from the full three dimensional finite model. A ¼ 0 I M 1 K M 1 C ; B ¼ " ( )# 0 0 M 1 ; E ¼ I " 0 ( MR )# M 1 R T MR þ M b (7) FIGURE 2 Placement of controllers

5 5of14 M, C, and K represent the combination of the parameters of superstructure and base level as follows. M ¼ M MR R T M R T ; C ¼ C MR þ M b 0 C b3 3 ; K ¼ K 0 0 K b ; u ¼ 0 : (8) F F contains the nonlinear force of bearing and control forces that is produced by the controllers. Using the Bouc Wen model for controllers at the base level causes the enveloped forces to be state dependent. For this reason, the fourthorder Runge Kutta method used for acquiring the nonlinear control forces. This type of Runge Kutta is one of the most commonly utilized kind of the Runge Kutta family that determines the response of the problem in every time step by using the initial value of problem. Also, Newmark's unconditionally stable constant average acceleration method was adopted in order to solve the state space equation. Additionally, the nonlinear force of smart base isolation system updated at each time step of earthquake record by using fourth order Runge Kutta method. Under certain circumstances, Runge Kutta method is utilized for convenience of solving the problem. Also, the pseudoforce method is used for determination of nonlinear force vector. One of the assets of this method is that coefficients of the method are determined only once for the constant time step, unlike other methods. Another benefit of using this technique is the proper performance in the highly nonlinear problems. Subsequently, Equation (6) can be solved from the trapezoidal rule, as follows: X kþ1 ¼ X k þ Δt ð 2 g k þ g kþ1 Þ; (9) where g kþ1 ¼ g X kþ1 ; u kþ1 ; U gkþ1 Þ: (10) In the above equation, g is the function of state space vector (Equation 6) that requires the iteration at every time step (k) to get the responses of the structure. In brief, the nonlinear forces in the smart base isolation system are obtained by solving the Equations 1 3 with using the Runge Kutta method, and after that, Equation (6) is resolved in a large number of iterations to attain the predetermined tolerance that it is The tolerance can ensure the dynamic stability of this method. 5 CONTROL STRATEGY To effective mitigation of the seismic responses of the benchmark structure under excitation of a given earthquake, an efficient control scheme is to be considered. In particular, in plane irregularity and three dimensional modeling of such structure may complicate the control procedure. The sliding mode control scheme enhanced with Fuzzy rules proposed as a control strategy. 5.1 Sliding mode control scheme In this paper, the sliding mode control method has employed for nonlinear control of the smart base isolated benchmark structure. As mentioned previously, base isolation system is considered such as a nonlinear system and modeled by Bouc Wen hysteretic model. This method was extended for utilizing in the three dimensional building, including coupled lateral torsional equations. The sliding mode control is a widely used method in classical control theory for structures with nonlinearity. Although the sliding mode control suffers from phenomena known as chattering, it provides an efficient tool to handle nonlinear systems. Chattering is an unsuitable oscillation with finite amplitude and frequency that presented by using discrete time sliding mode theory. There are several ways to diminishing this phenomenon; the natural way to decrease chattering phenomenon is the enhancement of switching frequency. Another way is assuming the amount of discontinuous control and reducing it to the level that needed for preserving sliding mode. Accordingly, we intended to eliminate the above disadvantages by utilizing a combination of sliding mode with Fuzzy rules. In the sliding mode control, a reaching surface, S, is defined. S ¼ PX (11) The reaching surface, S, contains r sliding variables (r is the total number of controllers). P is the vector of coefficients, and X is state space vector. With derivation of above equation and assuming S _ ¼ 0 for keeping the trajectory on the sliding surface and with using Equation (6), we can acquire equivalent control force: U eq ¼ ðpb u Þ 1 PAX: (12)

6 6of14 And we also obtain h i _X ¼ A B u ðpb u Þ 1 PA X: (13) By assuming the transformation matrix, D, the equation of motion is converted to a regular form, Y, as follows: Y ¼ DX; " # D ¼ I 2n 1 B 1 B I r ; (14) in which B 2 is a nonsingular matrix and Y is assumed into two parts that the first (2n r) rows are Y 1 and (r) rows are Y 2 vector, respectively. B ¼ B 1 B 2 (15) And consequently, _Y ¼ AY þ BU; (16) A ¼ DAD 1 ; P ¼ PD 1 ; B ¼ 0 B 2 ; (17) where A 11 ; A 22 ; P 1 ; P 2 are (2n r) (2n r), r r, r (2n r), r r, respectively. AssumingP 2 ¼ I r for simplicity and by substitution of the following equation into Equation (17), we have the transformed form of state space equation as follows: _Y 1 ¼ A 11 Y 1 þ A 12 Y 2 : (18) The matrix P 1 can be determined by using above equation. Thus, the sliding surface can be obtained. Several approaches can be used for determination of sliding surface; in this case, we use the linear quadratic regulator (LQR) method. The sliding surface can be achieved by minimizing the integral of a quadratic function of state vector that is described as following. J ¼ X ðþqx t ðþdt; t (19) 0 h i T J ¼ Y 1 ; Y 1 Y 2 dt: (20) 0 For the benchmark structure, the diagonal weighting matrix (Q) is given as follows: Q i = 80,000 for all DOFs of displacements except Q i = 8,000 for i =1,, 6 and Q i = 1 for all velocities elements. Finally, the sliding surface is designed as the following equation. Y 2 P ¼ PD ¼ P 1 I r D (21) In this paper, we design the controllers based on Lyapunov function, as follows: V ¼ 0:5S S ¼ 0:5X P PX: (22) With derivation of the equation and using the main state space equation, we have _V ¼ λðu GÞ ¼ r λ i ðu i G i Þ; (23) i¼1 where λ ¼ S PB; G ¼ ðpbþ 1 PAX ð Bf þ EÞ: (24)

7 7of14 With respect to _ V 0, the control force of sliding mode in the discontinuous form is obtained from the below equation: u i ðþ¼g t i δ i signðλ i Þ: (25) δ i is assumed 500 for all earthquake records, and δ is the diagonal matrix and (r r) dimensional. 5.2 Fuzzy logic In the recent years, fuzzy logic has recently obtained considerable attention in civil engineering. This logic has many advantages in comparison with classical mathematical theories. In general, fuzzy logic provides an approach to model uncertainty and imprecision. Also, this system utilizes only a linguistic description, and it does not require an analytical description. This system uses variables to simplify the relationship of inputs and output. According to the mentioned benefits, fuzzy logic is similar to engineering theories, because almost all of the complex engineering theories model the real behavior in an approximate manner. Therefore, fuzzy logic is one of the best theories that can deal appropriately with complex systems such as nonlinear control law, control law with uncertainties, and so on. For these reasons, we applied the fuzzy theory to control force equation for decreasing the chattering phenomena and optimizing the control force. For achieving these goals, the second part of control force equation that contains sign function was substituted with fuzzy logic. As usual, every fuzzy logic consists of three main steps: fuzzification, fuzzy interface engine, and defuzzification. In the first part of the fuzzy logic, inputs, and outputs determined by membership functions. The shape of the utilizing membership is a crucial criterion that depends on the variables. Also, in this process, different variables can take values in the interval [0, 1]. In this study, we used triangular membership function for inputs and output. In this case, λ and λ _ are considered as inputs of fuzzy logic that represents λ and changes of this variable. Also, control force is chosen as the output (Figure 3). The main part of the fuzzy logic is fuzzy interface engine that determines the relationship between inputs and FIGURE 3 (a) Membership function of input based on λ; (b) membership function of input based on λ.; (c) membership function of output based on control force (u)

8 8of14 output. This database demonstrates the collection of fuzzy IF THEN rules. This rule base can be determined based on the outputs. Table 1 represents the using rule base for this study. In the rule base table, positive, negative, and zero are abbreviated P, N, and Z, respectively. The final part of the fuzzy logic is defuzzification. In this part, the results of fuzzy logic change into crisp quantities due to further data processing. There are many defuzzification methods, but this paper uses the center of gravity method. 6 EARTHQUAKES In order to scrutinize the effects of earthquake excitation on the three dimensional benchmark building, seven earthquake records related to Newhall, Sylmar, El Centro, Rinaldi, Ji Ji, and Erzincan were chosen. The selected earthquake records have both the fault normal and fault parallel (FP) components. The characteristics of selected earthquake records, for instance, PGV and PGA, are displayed in Table 2. TABLE 1 Fuzzy rule base database λ _λ NB NM NS Z PS PM PB PB PM PS NS NM NM NM NB PM PB PS Z NS NS NM NB PS PB PM PS NS NS NM NB Z PB PM PS Z NS NM NB NS PB PM PS PS NS NM NB NM PB PM PS PS Z NS NB NB PB PM PM PM PS NS NM TABLE 2 The earthquake record Earthquake Station PGA(g) PGV (cm/s) Northridge USA, 1994 Sylmar Rinaldi Newhall Kobe Japan, 1995 Kjma Chi chi Taiwan, 1999 JiJi Erzincan Turkey, 1992 Erzincan Imperial Valley, USA, 1979 El Centro FIGURE 4 The displacement of eighth floor under the Sylmar (FP X)

9 9of14 7 RESULTS In this section, the responses of the benchmark building are evaluated by applying the proposed control theory under the seven earthquakes in the three dimensional structure. The displacement/rotation and velocity responses of the eighth floor under the FP component of the Sylmar earthquake have chosen to demonstrate. Time history responses are plotted in Figures 4, 5. It can be seen that the displacement and velocity responses of roof tend to decay by applying the considered control theory. For instance, the maximum displacement of the uncontrolled case is 0.35m, and the controlled case data are 0.05 m. Additionally, as expected, a significant reduction can be seen at the third DOF of the structure by 3D modeling of the entire building. Thus, structural responses in rotational DOF can be amended in such an irregular building. It should be noted that the third DOF is the torsional DOF and usually increases with the irregularity of structures. FIGURE 5 The velocity of eighth floor under the Sylmar (FP X) FIGURE 6 The displacement of base level under the Sylmar (FP X)

10 10 of 14 The results show that the velocity responses of the three dimensional building are decreased compared with the responses of the uncontrolled building. Also, the same results have obtained in the third DOF. Comparing the results of the uncontrolled system with the controlled system at the base level under earthquake excitations demonstrates the significant improvement in the base level responses. Figure 6 displays the result of the displacement/rotation response at the base level of the building, under FP component of the Sylmar earthquake. This plot shows that maximum displacement is 0.35 and 0.6 m in x and y directions, respectively. However, for the controlled case, maximum displacements of x and y directions are 0.02 m, approximately. Besides, the base level velocity under the Sylmar (FP X) is also shown in Figure 7. The results indicate that the proposed control strategy has reduced the velocity responses under specified earthquake ground motions. In control theory, one of the factors to investigate the performance of the control strategy is the control force. The conventional sliding mode control theory has a disadvantage called chattering phenomena. As discussed in previous sections, we utilized the fuzzy logic to repress and rectify this defect. Furthermore, this issue can lead to optimize the control force. Therefore, the control force for two kinds of strategies consists of FIGURE 7 The velocity of base level under the Sylmar (FP X) FIGURE 8 Control force El Centro (FP X)

11 11 of 14 conventional sliding mode control theory and fuzzy sliding mode control theory are determined in order to demonstrate the effectiveness of the proposed combination. The comparison is plotted in Figure 8. Besides, the maximum interstory drifts of the superstructure for the FP components of the earthquakes are shown in Figure 9. In this figure, drift is to be defined as the difference between displacements at two level of the structure. This graph shows that the response quantities are decreased substantially from the uncontrolled cases. FIGURE 9 Maximum interstory drift of superstructure under the (FP X) direction (x direction) FIGURE 10 Structural base shears and torsion moment of the building under the Newhall (FP X)

12 12 of 14 The base shear forces of the building in both controlled and uncontrolled cases are investigated. To show the performance of theory, we demonstrated the time history of base shear of the structure under FP component of Newhall earthquake in Figure 10. The figure illustrates time history of three dimensional structural shear contains two horizontal and a rotational DOF. The results show that using the proposed control strategy can lead to a considerable decrease in the base shears of the building. Also, the maximum value of torsion moment has been decreased by around a quarter. A significant reduction of base shear amounts especially in the third direction indicates that offered control theory can be controlled the three dimensional structure efficiently. As noted in Section 3, the nonlinear behavior of smart base isolation system is modeled by Bouc Wen model. For instance, Figure 11 illustrates the hysteresis loop demonstrated for Rinaldi earthquake (FP X). The similar results can be obtainable for other components of FIGURE 11 The hysteresis loops of nonlinear smart base isolation in x (East West) and y (North South) directions for Rinaldi Earthquake (FP X) TABLE 3 Performance indices Peak base shear Peak structure shear Peak base displacement J 1 ðqþ ¼ maxkv 0 ðt; qþk t; i maxbv i ðt; qþ t; i J 2 ðqþ ¼ maxkv 1 ðt; qþk t maxbv 1 ðt; qþ t J 3 ðqþ ¼ maxkd i ðt; qþk t; i maxbd i ðt; qþ Peak Interstorey drift Peak absolute floor acceleration Peak force generated J 4 ðqþ ¼ maxkd f ðt; qþk t; f maxbd f ðt; qþ t; f RMS Base displacement J 7 ðqþ ¼ maxkσ d ðt; qþk i maxkbσ d ðt; qþk i J 5 ðqþ ¼ maxka f ðt; qþk t; f maxkba f ðt; qþk t; f RMS absolute floor acceleration J 8 ðqþ ¼ maxkσ a ðt; qþk f maxkbσ a ðt; qþk f J 6 ðqþ ¼ t; i maxk F k ðt; qþk i maxkv 0 ðt; qþk Note. i = isolator number, 1,.,Ni (Ni = 8); k = device number, 1,..., Nd; f = floor number, 1,..., Nf; q = earthquake number; t = time, 0 < t < Tq; ( ) = inner product; = vector magnitude incorporating components of the earthquakes. t TABLE 4 Variation of performance of indexes Earthquake Case J1 J2 J3 J4 J5 J6 J7 J8 Newhall FP X FN X Sylmar FP X FN X El Centro FP X FN X Rinaldi FP X FN X Kobe FP X FN X JiJi FP X FN X Erzinkan FP X FN X

13 13 of 14 earthquakes. In order to assess the performance of proposed control theory, eight indices are assumed. The J 1 through J 8 indices measure the peak base shear, peak structure shear, peak base displacement, peak interstory, peak absolute floor acceleration, peak force generated, RMS base displacement, and RMS absolute floor acceleration, respectively. These indices are the proportion of controlled to uncontrolled cases that is described in Table 3. The values of the introduced indexes are obtained by using the offered control approach and under seven earthquake excitations. These values are demonstrated in Table 4. According to the results obtained in Table 4, the first index (J 1 ) that is related to base shear appropriately is decreased. Also, the values of third, fourth, fifth, and seventh indices in all earthquakes are appropriately decayed. Reduction in these indices demonstrates the efficiency of the proposed algorithm in displacements, drifts, and accelerations of the base and superstructure of the building. The amount of the sixth index shows that the usage energy for protecting the building is proportional to the parameters of earthquake excitation. The eighth index in almost all of earthquakes has a suitable diminution in acceleration. Finally, the amounts of indices are a clear cut issue of the usefulness of proposed control theory with consideration of rotational DOF. 8 CONCLUSION In this paper, fuzzy sliding mode algorithm was applied to control the nonlinear base isolated benchmark building under the earthquake excitation. The equation of motion comprises of base and superstructure equations was formed three dimensionally. The effect of rotational DOF was scrutinized in conjunction with two main DOFs in the model of structure, and control algorithm was implemented. The results showed that using the proposed strategy not only decrease the dynamic responses of the entire structure consists of acceleration, velocity, displacement, and rotation but also enhance sliding mode with fuzzy rules remove chattering in control forces created by actuators. Also, the result of numerical simulation of three dimensional benchmark building with base isolation system demonstrated that the isolation system response and superstructure responses appropriately decreased in controlled case. A significant decrease in the rotational response of the buildings at third DOF can be seen. The finding indicated the successful control of torsional responses in an irregular structure by using smart base isolation control scheme. ORCID Hosein Ghaffarzadeh T. Y. Yang REFERENCES [1] M. Mohebbi, H. Dadkhah, K. Shakeri, IJOCE. 2015, 5, 493. [2] J. M. Kelly, Earthquake Resistant Design with Rubber, in springer, [3] F. Naeim, J. M. Kelly, Earthquake Spectra 1999, 16,709. [4] J. M. Kelly, G. Leitmann, A. G. Soldatos, J. Optimiz. Theory App. 1987, 53, 159. [5] J. C. Ramallo, E. A. Johnson, J. B. F. Spencer, J. Eng. Mech. 2002, 128, [6] I. Buckle, S. Nagarajaiah, K. Ferrell, J. Struct. Eng Asce. 2002, 128, 3. [7] K. M. Choi, H. J. Jung, H. J. Lee, S. W. Cho, Struct. Control Hlth. 2008, 15, 758. [8] A. El Sinawi, R. Kashani, J. Vib. Control. 2001, 7, [9] Y. Ribakov, Struct. Des. Tall Spec. 2011, 20, 757. [10] C. Chang, B. F. Spencer, P. Shi, Struct. Control Hlth. 2014, 21,484. [11] H. Yoshioka, J. C. Ramallo, B. F. Spencer, J. Eng. Mech. 2002, 128, 540. [12] V. A. Matsagar, R. S. Jangid, J. Vib. Control. 2010, 16, [13] S. Narasimhan, S. Nagarajaiah, E. A. Johnson, H. P. Gavin, Struct. Control Hlth. 2006, 13, 573. [14] S. Suresh, S. Narasimhan, N. Sundararajan, Struct. Control Hlth. 2008, 15, 585. [15] S. Narasimhan, S. Nagarajaiah, E. A. Johnson, Struct. Control Hlth. 2008, 15, 657. [16] H. Ghaffarzadeh, E. A. Dehrod, N. Talebian, J. Vib. Control. 2012, 19, [17] H. Kim, P. N. Roschke, Eng. Struct. 2006, 28, 84. [18] X. Gu, J. Li, Y. Li, M. Askari, J. Intel. Mat. Syst. Str. 2016, 27, 849. [19] H. Alli, O. Yakut, Struct. Eng. Mech. 2007, 26, 517. [20] L. Li, G. Song, J. Ou. J. Vib. Control. 2010, 16, [21] C. M. Chang, B. F. Spencer, Struct. Control Hlth. 2010, 39, [22] S. Narasimhan, S. Nagarajaiah, Earthq. Eng. Struct. D. 2006, 35, 921. [23] N. C. Van Engelen, D. Konstantinidis, M. J. Tait, Earthq. Eng. Struct. D. 2016, 45, 421. [24] A. B. Habieb, G. Milani, T. Tavio, Eng. Fail. Anal. 2018, 90, 380. [25] D. Kar, R. Roy, J. Earthq. Eng. 2016, 2469,1.

14 14 of 14 [26] D. Pant, C. Wijeywickrema, Earthq. Eng. Struct. D. 2014, 43, [27] B. M. Constantinou, A. Member, A. Mokha, A. Reinhorn, J. Struct. Eng Asce. 1990, 116, 455. [28] S. Nagarajaiah, A. M. Reinhorn, M. C. Constantinou, J. Struct. Eng Asce. 2035, 1991, 117. AUTHOR BIOGRAPHIES Hosein Ghaffarzadeh is a professor in earthquake and structural engineering at the University of Tabriz. His research focuses on the structural control and structural nonlinear dynamics. Amir Hossein Ghaffari is a PhD student in Tabriz University, Tabriz, Iran. His current research interests include control of structures, structural seismic performance, and seismic design. T. Y. Yang is a professor in structural and earthquake engineering at the University of British Columbia. His research focuses on the performance assessment and design of structures under seismic loads. How to cite this article: Ghaffarzadeh H, Ghaffari AH, Yang TY. Fuzzy sliding mode control of nonlinear smart base isolated building under earthquake excitation. Struct Design Tall Spec Build. 2019;28:e