Microsoft Word - numero_69_art_07_4821.docx A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 89 Methodology to minimize the dynamic response of tall buildings under wind load controlled through semi-active magneto-rheological dampers Alex Koch de Almeida, Francisco da Silva Brandão Federal University of Rio Grande do Sul (UFRGS), Brazil. alex.almeida@ufrgs.br, https://orcid.org/0000-0001-9923-3434 eng.fsbrandao@gmail.com, https://orcid.org/0000-0001-7888-6321 Letícia Fleck Fadel Miguel Department of Mechanical Engineering (DEMEC), Federal University of Rio Grande do Sul (UFRGS), Brazil. letffm@ufrgs.br, https://orcid.org/0000-0001-9165-4306 KEYWORDS. Vibration Control, MR dampers, Structural Optimization, Tall Buildings, Wind Load. Citation: Almeida, A. K., Brandão, F. S., Miguel, L. F. F., Methodology to minimize the dynamic response of tall buildings under wind load controlled through semi-active magneto-rheological dampers, Frattura ed Integrità Strutturale, 69 (2024) 89-105. Received: 26.01.2024 Accepted: 15.04.2021 Published: 25.04.2024 Issue: 07.2024 Copyright: © 2024 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. INTRODUCTION he construction of tall buildings has become easier, mainly due to technological advances in materials and construction techniques. Linked to the growth of urban centers, which causes a decrease in free spaces for new constructions, building taller structures has become a necessity. Consequently, buildings are being built increasingly taller and slender, and therefore more vulnerable to wind action [1] Thus, an important area of research is the study of slender structures, such as tall buildings, subject to the dynamic effects of wind. It is essential to design structures that meet comfort and safety requirements at the lowest possible cost. In this context, the area of structural optimization together with the area of vibration control becomes fundamental. Regarding the area of optimization, both deterministic and stochastic methods can be used [2], with an advantage for stochastic methods when dealing with complex problems such as the one studied in the present work. Several topics are T https://youtu.be/Eno_uKaICe8 A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 90 currently studied in the field of structural mass optimization, for example, studies on minimizing the mass of structures subject to natural frequency constraints [3-7]. Although structural optimization is an effective tool, the combination of this with other tools is often necessary. Considering this context, the vibration control devices, classified as passive, active, and semi-active can be added to the structures to improve their performance against dynamic actions, and the optimization process can be also implemented to reach the maximum performance of these devices. The passive devices, which have pre-defined properties, are characterized by not using an external power source, while the active ones, which apply force to the structure at the same time as the excitation, need an external power source. The semi-active devices, which have been the object of several recent research, combine the advantages of passive and active devices simultaneously as they have an intrinsic characteristic of adaptability, being able to change their properties with a reduced amount of energy, without applying force to the structure. Among the semi-active control devices, the Magneto-Rheological (MR) dampers stand out, due to their mechanical simplicity, wide dynamic applicability, low energy cost, great strength, and robustness. These characteristics have shown good adherence to the demands of structural systems in the control of dynamic excitations such as earthquakes and wind [8]. Therefore, civil engineering can take advantage of this type of approach, creating structures able to monitor and control their response under dynamic excitations. Several studies have been published demonstrating the application of MR dampers in the control of dynamic responses. Among the first experimental studies with prototypes, there are those in which the authors developed experiments with structures of up to 6 degrees of freedom excited at the base in order to simulate an earthquake [9-11]. There are also numerical studies with simple structural models in which the authors developed simulations with structures of up to 20 degrees of freedom excited by earthquakes [12-15], and a study in which the authors developed simulations with a structure of 40 degrees of freedom excited by the wind [16]. It is also worth mentioning the more complex numerical models, such as the one in which the authors studied a structural model called mega-sub-controlled structure excited by wind [17] and the one in which the authors performed a simulation with a 2D frame excited by earthquakes [18]. Hybrid strategies using MR dampers and other types of devices are also being researched, for example in [19]. Finally, considering the theoretical- experimental studies, there is a study in which the authors reported an experiment that was carried out on the cable-stayed Dongting Lake Bridge in China, severely excited by strong winds and rain [20], and the one in which the author analyzed several different structures, considering numerical models with multiple degrees of freedom excited by earthquakes [21]. As highlighted, most of these studies focus on analyzing simplified structural models subjected to earthquakes and, therefore, there is a lack of studies with more complex structural models, able to provide a better description of the behavior of tall buildings subjected to wind excitation. Additionally, unlike most works in the literature, this paper proposes a methodology in which not only the installation of dampers is considered as a way of minimizing vibration amplitudes, but also the optimization of the structure is carried out, with the objective of increasing the fundamental frequency of the building, taking it to values further away from the frequency content of the wind spectrum, and thus reducing the dynamic response. That is, the proposed methodology combines structural optimization with semi-active control devices. Besides, most of the works that propose different types of vibration control systems, for different types of structures and excitations, are related to passive systems, for example [22-42, among others]. Thus, to contribute to filling these gaps, this study focuses on analyzing a tall building, described by a 2D frame model of multiple degrees of freedom, under dynamic wind loading and controlled by semi-active MR dampers. For this, the dynamic responses of three different configurations of the building, under wind excitation, are analyzed, and compared with performance criteria indicated in [43] and [44]. The first configuration, called Original Uncontrolled (C1), consists of a frame building extracted from a tall building originally proposed and analyzed in [45]. The second, called Optimized Uncontrolled (C2), consists of a structure whose fundamental frequency was optimized, via the PSO algorithm, as a function of its mass, from the C1 configuration. The complete procedure for this optimization was presented by the authors in a previous paper [46]. The third and last one, called Optimized Controlled (C3), consists of the C2 configuration controlled through a set of semi-active MR dampers. PROBLEM FORMULATION AND PROPOSED METHODOLOGY Structural modelling he numerical modelling of the structure is approached through the finite element method, considering a 2D frame model, according to the procedures from [47]. The damping matrix, C , was generated using the Rayleigh method, formulated as a linear combination of the global mass matrix, M , and global stiffness matrix, K . T A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 91 Wind modeling For the proposed problem, the procedures described in [48] are followed and, therefore, it deals only with synoptic winds (more complex models can be found in [49]). Thus, the wind load is given by:  D D DF F F  (1) in which DF is the mean component and DF is the fluctuating component of the drag force, DF . The mean component of the drag force can be obtained by:  22 0 / p D D i i rF q C A b z z (2) in which DC indicates the drag coefficient that depends on the building shape, iA is the effective area of exposure considered, orthogonal to the wind direction,  b and p are meteorological parameters, rz is the reference height (10 meters), iz is the height under analysis and 0q is the reference dynamic pressure of the wind, relative to the mean component, given by: 2 0 1 2 a pq V (3) in which a represents the specific mass of the air (equal to 1.225 kg/m³ at 15 ºC and 1013 mbar) and pV is the design wind velocity, expressed by: 0 1 30.69pV V S S (4) in which 0V is the base wind velocity, 1S is the topographic correction factor and 3S is the statistical correction factor. The fluctuating component of the drag force can be obtained by:   0 D iDF q C A (5) in which 0q is the reference dynamic pressure of the wind, relative to the fluctuating component, given by:   0 1   2 / 2 p a i rq Vpb z z v  (6) in which  x y  c ,c ,v t is the fluctuating component of the wind velocity, xc and yc are the horizontal and vertical coordinates, respectively, in a Cartesian plane, of the point under analysis, and t is the time. The fluctuating component of wind velocity is considered a normal random process with zero mean. The problem was formulated through the superposition of harmonic waves, in a process known as the spectral representation method [50]. Using this method, it is possible to convert the energy described by the spectrum in the frequency domain to the time domain and this implies the inclusion of a random component in the process, as shown in:       1   2  cos 2 Φ fn s i i i i i i V t S f f f t      (7) in which  sV t is a fluctuating velocity signal at a given position in space, with  1, 2,3,4s  , Si is the spectral density of the wind velocity, if is the frequencies considered, fn is the maximum value of the considered frequency range, if is A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 92 the frequency increment and Φi is the phase angle which is a random variable with a uniform probability distribution function between 0 and 2𝜋.Among the available spectral models, the one proposed by Davenport is used, according to [51], described by:     2 * 2 4/32 * *    4    1 i i if S f n u n   ; * 10 * 0 ln ref k V u z z        ; * * 10 if L n V  (8) in which *u is the friction velocity, *n is the dimensionless frequency, *k represents the Kármán constant, 10V is the mean wind velocity at 10 meters above ground level, refz is the reference height, oz is the roughness length and *L is a fitting constant of the spectral model. In order to consider the spatial correlation among the signals of fluctuating velocity, the fluctuating component of the wind velocity is determined, according to [52] as a result of the approach proposed in [53], through:                     2 1 3 1 1 4 3 2 1  Δ Δ  Δ Δ Δ , ,    Δ  Δ Δ Δ Δ x y x y a b x y a b V t V t V t V t V c c t V t c c c c V t V t V t V t c c c c           (9) Eqn. (9) determines the fluctuating velocity at a given point of interest from horizontal and vertical coordinates, xc and yc , respectively, in a Cartesian plane, where the s fluctuating velocity signals are spaced by a horizontal correlation length ac and by a vertical correlation length bc . Therefore, the frame under study is inserted perpendicularly into the correlation plane, and the fluctuating velocity is determined at each of the external nodes of the structure. ac and bc are determined through Eqn. (10) proposed in [52] as a result of linear regression applied to experimental data from [51], where cz is the structure height. 1.60 22.1 0.93 29.3 a c b c c z c z     (10) MR damper modeling The semi-active control system usually is originated from passive control systems that are modified to allow adjustment of mechanical properties, for example, devices that dissipate energy through modified viscous fluids to behave in a semi-active configuration. On the one hand, as in an active control system, sensors installed in the structure monitor the response, and a controller, based on the response, generates an appropriate command signal for the device. On the other hand, as in a passive control system, the control forces are developed as a result of the movement of the structure itself [54], that is, semi- active dampers have mechanical properties or parameters that can be adjusted to improve their performance as an active control system, maintaining the reliability of passive control systems [55]. Among the semi-active control devices, the controllable fluid dampers stand out, employing fluids in their interior that can adjust their mechanical properties quickly in reaction to external forces. Among the applicable fluids, the magneto-rheological (MR), like the model shown in Fig. 1, has as an essential feature its ability to reversibly change from a free-flowing linear viscous fluid to a semi-solid with a controllable flow force, in milliseconds, when exposed to a magnetic field [56]. MR dampers consist of a cylinder that contains the MR fluid, manipulated through a diaphragm and excited by a coil responsible for transmitting the magnetic signal that changes its properties. The reactive force is transmitted by the Only moving part, the piston. Fig. 1 shows the schematic of the components of an MR damper. These devices are simple to operate and maintain, have high reliability, and are stable over a wide temperature range. Since they are basically adaptive passive devices, even in case of a malfunction of their semi-active property, the controller in the passive configuration can still contribute to mitigating the effects of dynamic actions. Furthermore, given their high strength, A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 93 they are capable of being applied to civil structures using a small energy supply. Under these conditions, MR dampers are promising devices for application in structures subjected to dynamic wind actions. Many rheological models have been developed to describe the behavior of these devices. In this study, the model proposed by Bouc-Wen and adjusted by [55], named modified Bouc-Wen, is used, as shown in Fig. 2. Figure 1: Schematic of the components of an MR damper. Figure 2: Modified Bouc-Wen rheological model. This model is governed by:  1 1 0MRF c y k x x   (11)  0 0 0 1 1 y z c x k x y c c         (12)    1bw bwn n bwz x y z z x y z A x y              (13) in which  MRF t is the total reactive force generated by the system, 0c is the viscous damping observed at higher velocities, 0k is present to control stiffness at higher velocities, 1c is a damping included in the model to produce the roll-off effect observed at low velocities, 1k is the stiffness of the accumulator, x and y are the displacements of the damper, z is the evolutionary variable, 0x is the initial displacement of the spring 1k associated with the nominal force of the accumulator.  ,  ,  ,   bwA e bwn are parameters that describe the hysteresis of the system. The model parameters that are used in this A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 94 work were experimentally determined by [21] for the MR RD-1005-3 damper (Lord corporation) and are presented in Tab. 1, where I is the electrical current supplied to the device, whose maximum value, associated with the saturation of the magnetic field, is 0.5 A. Independent parameters  bwA  1mm    1mm     0 /k N mm   1 0 Nk x x bwn 10.013 3.044 0.103 1.121 40 2 Current dependent parameters    3 2I 826.67I 905.14I 412.52I 38.24  N         3 2 0c I 11.73I 10.51I 11.02I 0.59  N.s / mm        3 2 1c I 54.40I 57.03I 64.57I 4.73  N.s / mm     Table 1: Parameters of the modified Bouc-Wen model for the MR RD-1005-3 damper [21]. The MR damper RD-1005-3 is used for low force capacity applications, such as industrial suspensions. It can be verified in its specifications that its peak force in response to a current value of 1 A is 2,224 kN [21]. Knowing this, the total reactive force of the equipment ( MRF ) will be multiplied by an amplification factor (Ω ). This adjustment represents Ω dampers acting in parallel on each m controlled mass, in order to simulate a robust device compatible with the drag force. Equation of motion The numerical modeling of the system considering the damping forces of the MR dampers was approached according to [57], in this way, the matrix representation of the dynamic equilibrium equation is: ¨ Ω MRM x C x Kx F F        (14) in which x  , x   , and x   are the displacement, velocity, and acceleration vectors, respectively. F  is the external forces vector and MRF  is the damping forces vector, both applied at the indicated degrees of freedom. To solve Eqn. (14) in the time domain the Newmark method associated with the Runge-Kutta (RK4) is applied. For C3, the dynamic equilibrium equation is solved repeatedly, gradually increasing the amplification factor Ω until satisfying the performance criterion, according to the pseudocode shown in Fig. 3. Figure 3: Pseudocode for defining the amplification factor. LQR-CO control strategy Active structural control research efforts have focused on a variety of control techniques based on several design criteria. Some are considered classic, as they are direct applications of modern control theory, among them is the optimal control technique [58]. As highlighted in [59], the optimal controller problem can be defined as the determination of a control law for a given system in order to reach a specific optimal criterion by minimizing a pre-defined performance index. Among the strategies associated with optimal control, there is the so-called clipped optimal (CO), developed in [9], which, according to [60], is the most successful strategy so far for the control of systems that use controllable fluid devices. In this case, a A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 95 controller is designed based on linear control strategies, such as the Linear Quadratic Regulator (LQR), for example, as if the control device were active. However, a decision block for the current applied to the actuator and measuring the actuating control force are integrated into the system to properly adjust the control command and accommodate dissipative characteristics and non-linearities in device behavior. The LQR, widely studied and widespread, can be established as a control engineering tool that aims to determine an ideal control by minimizing a quadratic performance index when the control is a linear function of the response [58, 59, and 61]. To take advantage of the semi-active behavior of MR dampers, the LQR controller associated with the Clipped Optimal (LQR-CO) strategy is used. Fig. 4 shows the block diagram of this strategy. Figure 4: LQR-CO Block Diagram. The LQR-CO strategy works through the cycle shown in Fig. 4, as follows:  The structure is excited by an external force (input);  The structure reacts to the excitation and presents a response (output), in terms of displacement, velocity, and acceleration;  The response is captured by sensors installed in the structure (or in numerical simulation it is obtained by integration) that take this information to the controller;  Based on the response, the LQR controller determines an optimal control force and sends this information to the current decision block;  The current decision block compares the actuating control forces with those determined by the controller and then decides the current to be applied to the actuators in order to position the system control forces as close as possible to the optimal forces defined by the controller;  Since actuators have their properties controlled by the current, a new current produces new control forces that are applied to the structure. The optimal control force at each instant of time can be determined by Eqn. (15), approached by [58, 59, and 61].    1 0 1 2 Tf t R B Pe t    (15) 11 2 0 2 T TPA PBR B A P Q    (16)    , , 1 1 0 n n n nId A M K M C           (17)     , 1 , 0 n n n m B M          (18)       n n x t e t x t          (19) A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 96 in which  of t  is the vector of optimal forces at each instant of time, with the m optimal forces of and n control forces applied to the system.  e t  is the state vector of the system composed of the n displacements,  x t , and the n velocities,  x t , and finally, n is the number of degrees of freedom of the system. B is the matrix that describes the control forces in the state space,  is the matrix that describes the location of the m control forces. Q and R are called weighting matrices, high values for elements of Q mean prioritizing the reduction of the response over control forces, and high values for elements of R mean the opposite; in general, these values are obtained in a testing process aiming at the best result. A is the system state matrix and Id is the identity matrix. P is the Riccati matrix. Once the vector  of t  is determined, the selection of the current to be applied to the damper can be obtained, according to [9], by:  max o mr mrI I H f F F    (20) in which maxI is the maximum current associated with the saturation of the magnetic field and     H  is the Heaviside function. In this way, the damper force is controlled indirectly, through current control, that is, when the damper is providing the optimal force, the applied current remains unchanged, if the magnitude of the force produced by the damper is less than the magnitude of the desired optimum force and both forces have the same sign, the applied current is increased to the maximum level. Performance criteria Three different performance criteria, related to the Serviceability Limit State, are considered in this paper. The first, indicated in Appendix CC of [43], refers to the maximum permissible horizontal displacement of the building ( maxD ), determined through Eqn. (21), in which tH is the total height of the building. 600 t max H D  (21) The second indicates the maximum permissible displacement between adjacent floors (story drift). According to the American standard [43], the story drift (SD) cannot exceed approximately 1 cm, in this case 1maxSD cm . Both limits are generally sufficient to minimize damage to the wall covering and non-load-bearing walls [43]. Finally, the third is related to the maximum permissible acceleration ( maxAcc ). According to [43], continuous vibrations (over a period of minutes) with an acceleration of the order of 0.005  g to 0.01g , in which g is the acceleration due to gravity, are uncomfortable for most people. Tab. 2 presents the acceleration limits related to user sensitivity from [44]. Perception maxAcc Imperceptible < 0.005 g Noticeable 0.005 g to 0.015 g Uncomfortable 0.015 g to 0.05 g Very uncomfortable 0.05 g to 0.15 g Intolerable > 0.15 g Table 2: Limit acceleration according to [44]. It is important to note that this is not the main design criterion because, depending on the recurrence time, these accelerations, even if uncomfortable, are acceptable [44]. Thus, considering the previous information, maxAcc 0.01g is adopted as the limiting criterion, thus allowing accelerations within the noticeable sensitivity range. A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 97 RESULTS AND DISCUSSIONS Analyzed structures hree different structural configurations are analyzed in this study. The first, C1, is characterized by a 2D frame extracted from a building previously proposed and analyzed by [45] designed as a reinforced concrete structure with fck equal to 50 MPa, symmetrical and with dimensions in plan of 15m × 15m, 35 floors, 99.75m high, 2.85m high between slabs and 12cm thick on each slab. The building analyzed in this work is considered fixed at the foundation and composed of rectangular columns on the sides, “L” columns in the central part, and continuous rectangular beams discretized in 144 nodes, 245 elements, and 432 degrees of freedom. The details of this frame in its Original Uncontrolled configuration (C1) are shown in Fig. 5. In this configuration, the dimensions of the elements are repeated on all floors. Figure 5: Structure analyzed in uncontrolled original configuration (C1). a) 2D frame, b) Cross section in plan considered, c) Cross section of columns (E1 and E2) and beams (E3 and E4). At the mass matrix, E and  are taken as 103.4  x10 N/m² and 2500 kg/m³, respectively, and concentrated masses of the slabs are considered at the respective nodes of each floor, considering the appropriate influence area. Thus, the structure in its Original Uncontrolled configuration (C1) has a fundamental frequency of 0.34 Hz and a mass of 1171.6 tons. The second configuration, called Optimized Uncontrolled (C2), consists of the C1 configuration in which the fundamental frequency is optimized, via the PSO algorithm, as a function of its mass. That is, the objective function of the optimization process, which has the maximum mass as a constraint, is to maximize the fundamental frequency of the building, taking it to values further away from the frequency content of the wind spectrum, and thus reducing the dynamic response. This complete optimization procedure was presented by the authors in a previous work [46]. In this scenario, the dimensions of the elements vary depending on their position, and their values can be verified in the authors' previous paper [46]. Thus, the structure in the Optimized Uncontrolled configuration (C2) has a fundamental frequency of 0.50 Hz and a mass of 1526.7 tons, which means an increase of almost 50% (47.06%) in the fundamental frequency against an increase of approximately 30% (30.31%) in the mass of the structure in relation to the C1 configuration. The third and last one, called Optimized Controlled (C3), presents the C2 configuration with the application of a set of semi-active MR dampers, detailed in the next section. T A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 98 Vibration control through semi-active MR dampers In order to obtain the Optimized Controlled configuration (C3), 35 MR dampers are applied to the central frames of the Optimized Uncontrolled structure (C2), one per floor. Each floor is controlled by the damping force MRFm , with  1   35m to , as illustrated in Fig. 6. Figure 6: Position of the dampers. In the LQR solution, the following weight matrices are considered, as indicated by [62]:       , , , 0 0 0 n n n n n n K Q         and   7 ,10 d m mR I (22) Dynamic wind load The dynamic wind load is modeled according to the procedures presented previously. Tab. 3 presents a summary of the parameters used, in which 3S is adopted considering the Serviceability Limit State, with a recurrence time of 10 years. Parameter Unity Value DC - 1.45 iA m² 10.69 0V m/s 43 1S - 1 3S - 0.78 b - 1 p - 0.15 if Hz 510 ~ 10 *k - 0.4 refz m 10 oz m 0.07 *L m 1200 Table 3: Parameters used to determine the wind load. Fig. 7 shows the drag force at floor 01 (first), floor 20 (approximately half the height of the frame), and floor 35 (last floor). A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 99 Figure 7: Drag force. The three configurations, C1, C2, and C3, are horizontally subjected to the dynamic wind action, as shown in Fig. 8, and the dynamic equilibrium equation is solved for each of them. For C3 the amplification factor Ω that satisfied the performance criterion is 660. Figure 8: Structure analyzed. a) Perspective, b) 2D frame. Results of the structural configurations The first result to be analyzed refers to the optimization process of the structural elements. In this process, from the C1 configuration, the C2 configuration is obtained. It is possible to note an increase of 47.06% in the fundamental frequency at a cost of a 30.31% increase in the total mass of the structure. Next, the analysis of the three configurations C1, C2, and C3, subjected to dynamic wind loading is presented. The analysis is based on observing the response, in terms of displacement, story drift, and acceleration, over time (300s). The maximum displacement at each floor is shown in Fig. 9(a) and the maximum acceleration at each floor is reported in Fig. 9(b). Analyzing these figures, it can be seen that C3 is the best control scenario and presents considerable response reductions. At the top floor, the displacement is reduced from 16.20cm to 8.45cm considering the C2 configuration, and to 4.66cm, considering the C3 configuration. The maximum accelerations are 40.1cm/s², 41.2cm/s² and 9.36cm/s², in configurations C1, C2, and C3, respectively. The displacement over time at the top floor is shown in Fig. 10 and the acceleration is in Fig. 11. For the maximum story drift in each inter-floor, its representation is shown in Fig. 12, in which the maximum values are: 0.58cm, 0.32cm, and 0.23cm for C1, C2, and C3 configurations, respectively. A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 100 Figure 9: a) Maximum displacements and b) maximum accelerations per floor. Figure 10: Displacement over time at the top floor. Figure 11: Acceleration over time at the top floor. A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 101 Figure 12: Maximum story drift. Thus, it could be seen that, in terms of displacements, scenarios C2 and C3 showed a reduction of 47.8% and 71.2%, respectively, in relation to C1, and C3 presented a reduction of 44.85% in relation to C2. For the accelerations, C2 showed an increase of 2.7% and C3 a decrease of 76.7%, respectively, in relation to C1, and C3 reduction of 77.3% in relation to C2. Finally, in terms of story drifts, C2 and C3 showed a reduction of 44.8% and 60.3%, respectively, in relation to C1, and C3 reduction of 28.13% in relation to C2. For the building under study, the performance criterion related to the maximum horizontal displacement is 16.6  .maxD cm Thus, C1, C2, and C3 met the criterion. Regarding the performance criterion related to maximum story drift, ASCE/SEI 7- 16 [43] does not set a value, but indicates an approximation, 1maxSD cm . In this case, C1, C2, and C3 met the criterion. It is observed, therefore, that the three configurations met the criteria related to maximum horizontal displacement and maximum story drift. Considering that the maximum acceleration of configuration C3 occurs on the twenty-eighth floor and that this value is - 9.36m/s2, Fig. 13 shows the acceleration over time, for the C3 on the 28th floor, against the limits presented previously. Figure 13: Acceleration over time, C3 on the 28th floor, against performance criterion. Thus, regarding the performance criterion related to the user's perception, it was verified that on the 28th floor, critical in relation to the acceleration for the C3 configuration, the accelerations were below 0.01g , that is, the perception is noticeable. Therefore, it can be seen from Figs. 9b), 11, and 13 that only the C3 configuration met the criterion. It is observed, therefore, that in the C3 configuration, the accelerations over time were controlled. A. Almeida et alii, Frattura ed Integrità Strutturale, 69 (2024) 89-105; DOI: 10.3221/IGF-ESIS.69.07 102 CONCLUSIONS his paper proposed a new methodology to evaluate and optimize the dynamic behavior of tall buildings under wind loading controlled through semi-active Magneto-Rheological dampers. Thus, through this numerical study, it was possible to minimize the dynamic response of a tall building, described through a 2D frame model with multiple degrees of freedom. For this, the responses of three structural configurations (C1, C2, and C3) were evaluated in terms of displacement, story drift, and acceleration, with an evaluation of performance criteria indicated in the literature. For the maximum displacements at the top floor, it was found that C2 and C3 showed a reduction of 47.8% and 71.2%, respectively, in relation to C1, and C3 reduction of 44.85% in relation to C2. For the maximum story drift, it was verified reductions of 44.8% and 60.3%, to C2 and C3, respectively, in relation to C1 and C3 showed a reduction of 28.13% in relation to C2. Finally, for the maximum accelerations, C2 showed an increase of 2.7% and C3 a decrease of 76.7% in relation to C1, and C3 presented a reduction of 77.3% in relation to C2. Regarding the performance criterion by user's perception, it was found that the 28th floor (critical floor) presented perception noticeable, since the acceleration was below 0.01g . From the evaluation of the responses of the three configurations, it was found that the C3 configuration was the only one able to meet all the established performance criteria, and part of this behavior was due to the fundamental frequency optimization and another part was due to the influence of the MR dampers. Thus, the proposed methodology combining structural optimization and MR dampers proved to be a powerful tool for vibration control and it could be used to help designers of this type of structure. ACKNOWLEDGMENTS he authors acknowledge the financial support of Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil. REFERENCES [1] Tamura, Y. and Kareem A. (2013). Advanced structural wind engineering, Japan, Springer. [2] Saka, M.P. and Geem, Z.W. (2013). Mathematical and Metaheuristic Applications in Design Optimization of Steel Frame Structures: An Extensive Review, Mathematical Problems in Engineering, ID 271031. DOI: 10.1155/2013/271031. [3] Miguel, L.F.F. and Fadel Miguel, L.F. (2012). Shape and Size Optimization of Truss Structures Considering Dynamic Constraints through Modern Metaheuristic Algorithms, Expert Systems with Applications 39(10), pp. 9458-9467. 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Controle Semi-Ativo De Vibrações Em Estruturas Utilizando Amortecedor Magnetorreológico, PhD Thesis, Universidade de Brasília, Brasil. << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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