Re-centering variable friction device for vibration control of structures subjected to near-field earthquakes

Mechanical Systems and Signal Processing 25 (2011) 2849–2862

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Re-centering variable friction device for vibration control of structures subjected to near-field earthquakes Osman E. Ozbulut, Stefan Hurlebaus n Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843, USA

a r t i c l e i n f o

abstract

Article history: Received 16 November 2010 Received in revised form 20 April 2011 Accepted 30 April 2011 Available online 23 May 2011

This paper proposes a re-centering variable friction device (RVFD) for control of civil structures subjected to near-field earthquakes. The proposed hybrid device has two sub-components. The first sub-component of this hybrid device consists of shape memory alloy (SMA) wires that exhibit a unique hysteretic behavior and full recovery following post-transformation deformations. The second sub-component of the hybrid device consists of variable friction damper (VFD) that can be intelligently controlled for adaptive semi-active behavior via modulation of its voltage level. In general, installed SMA devices have the ability to re-center structures at the end of the motion and VFDs can increase the energy dissipation capacity of structures. The full realization of these devices into a singular, hybrid form which complements the performance of each device is investigated in this study. A neuro-fuzzy model is used to capture rate- and temperature-dependent nonlinear behavior of the SMA components of the hybrid device. An optimal fuzzy logic controller (FLC) is developed to modulate voltage level of VFDs for favorable performance in a RVFD hybrid application. To obtain optimal controllers for concurrent mitigation of displacement and acceleration responses, tuning of governing fuzzy rules is conducted by a multi-objective heuristic optimization. Then, numerical simulation of a multi-story building is conducted to evaluate the performance of the hybrid device. Results show that a re-centering variable friction device modulated with a fuzzy logic control strategy can effectively reduce structural deformations without increasing acceleration response during near-field earthquakes. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Shape memory alloy Superelasticity Seismic control Variable friction damper Near-field earthquake

1. Introduction In recent years, the damaging effects of near-field motions on civil structures have revealed the lack of conventional design methods and emphasized the need of innovative design strategies [1–3]. Numerous structures were damaged or collapsed during the 1994 Northridge, 1995 Kobe, 1999 Duzce, 1999 Chi-Chi and the most recent 2008 Winchuan earthquakes, leading not only to significant economic losses but also large loss of lives. Near-field earthquakes have longduration pulses with peak velocities of the order of 0.5 m/s. The ground motions with such velocity pulses may cause significant damage to structures within the near-field region. Another characteristic of the near-field motions that adversely influences structural response is that the ground motion normal to the fault trace is richer than that parallel to the fault in long-period spectral components. Due to this normal component of the near-field motions, large displacement demands are imposed on to the structures.

n

Corresponding author. Tel.: þ1 979 845 9570; fax: þ1 979 845 6554. E-mail address: [email protected] (S. Hurlebaus).

0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.04.017

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A number of researchers have investigated the effects of near-field ground motions on the dynamic behavior of structures [4–6]. Malhotra [7] studied the response of buildings to near-field pulse like ground motions and observed that pulse-like ground motions with high peak ground velocity to peak ground acceleration (PGV/PGA) ratio have several adverse effects such as increased base shear and inter-storey drifts in high-rise buildings, reduced effectiveness of supplemental damping and increased ductility demand on the response of the structures. Liao et al. [8] compared the dynamic behavior of a reinforced concrete building subjected to near-field and far-field ground motions. They found out that near-field earthquakes impose higher ductility and base shear demands than far-field earthquakes. In another study, Loh et al. [9] examined the effect of 1999 Chi-Chi earthquake on the response of bridge structures. Through numerical analyses and field investigations, they revealed that the near-field effects considerably amplify the response of bridges. Shape memory alloys (SMAs) have attracted a great deal of attention as a smart material that can be used in seismic protection systems for energy dissipating and re-centering purposes [10–19]. SMAs behave similarly to linear-elastic materials for small magnitude events, but for moderate and more severe strain levels, SMAs display superelastic behavior from which it can fully recover its original elastic shape. SMAs also exhibit self-centering behavior when permanent deformations in surrounding assemblies afflict the SMA installation; thus, the overall integrity of neighboring structural systems can be maintained. Because of their re-centering capability, SMAs can serve as a valuable component in a seismic control device. Although the hysteretic behavior of SMAs provide some level of damping, the quantity of equivalent viscous damping provided by superelastic SMA wires or bars is not sufficient to render the use of SMAs as the sole damping device implemented in a tall structure subjected to severe dynamic loadings [20,21]. Friction dampers rely on the resistance developed between two solid interfaces sliding relative to each other [22]. They add energy dissipation capacity to structures and limit the force and acceleration in the system, improving their ability to withstand extreme events, but they lack re-centering capacity [23]. The energy dissipated by a friction damper is the product of normal contact force and drift of the damper. Hence, for a friction damper, if a small normal force is available, the energy dissipation of the damper may not be substantial during severe earthquakes. On the other hand, a large normal force results in small deformation of the damper (i.e., less energy dissipation) during moderate and weak ground motions. Therefore, a controllable normal force is favorable in order to ensure sufficient energy dissipation for various levels of ground motions [24]. This paper proposes a novel re-centering variable friction device (RVFD) for control of civil structures that are subjected to near-field earthquakes. The first sub-component of this hybrid device consists of superelastic SMA wires that exhibit excellent re-centering characteristics but have limited energy dissipation capability. The second component consists of a variable friction damper (VFD) that can be intelligently controlled to enable desired level of energy dissipation through friction. Here, first, a detailed description of the hybrid device and its modeling technique are discussed. Then, a GA-based optimal fuzzy controller that is employed to modulate the normal contact force of the RVFD is introduced. Next, a numerical study is performed to investigate the effectiveness of the proposed device in decreasing the response of a threestory structure subjected to near-field earthquakes. 2. Re-centering variable friction device In this study, two different designs of a RVFD are proposed. The RVFD in both configurations consists of three parts: (i) a friction generation unit, (ii) a piezoelectric actuator, and (iii) shape memory alloy wires. Below is a brief description of each design concept. Fig. 1 shows a schematic diagram and a 3D rendering of the proposed RVFD in the Configuration I. In this design of the hybrid device, friction unit simply consists of two steel plates rubbing against a friction pad material and clamped together with high strength bolts and a piezoelectric actuator. Since piezoelectric materials cannot endure large strains that can be induced in the hybrid device during a seismic event, piezoelectric actuators are oriented perpendicular to the sliding surface of a RVFD. Four piezoelectric actuators are used in the device. The clamping force of a RVFD can be adjusted according to the current level of ground motion by adjusting the voltage level of piezoelectric actuators. The energy dissipation capacity of the hybrid device is a function of the friction material chosen and the contact force. For a given structural system subjected to seismic loading, the desired level of energy dissipation can be obtained by a proper selection of the coefficient of friction between two bodies and varying the contact force of the device. Superelastic shape memory alloys are employed in the hybrid device to realize re-centering ability for the device. A total of five studs are inserted into plates as shown in Fig. 1. Studs 1 and 3 are inserted into the inner steel plate and mutually move with this plate. In order to enable the movement of stud 3, the outer plate has a longitudinal slot in the middle of its top and bottom surfaces. Stud 4 is similarly inserted into the outer plate, while two short studs (2 and 5) are attached to the edges of the outer plate. The SMA wires are wrapped around the studs connected to the inner and outer plates. The arrangement of the wires and studs is such that either the wires in the middle group or the outer SMA wire groups are subjected to the tension. By setting the number of SMA wires in the middle group to twice as many as in the outer group, the hybrid device will exhibit symmetrical response. The schematic diagram of the RVFD in the Configuration II is shown in Fig. 2. A 3D rendering of the device is included in the figure. The functioning principle of the device is the same as the configuration I. Here, instead of flat steel plates, a sliding pipe-in-pipe system, where the inner pipe can slide within the outer pipe is used as friction generation unit. The pipes are held together with the help of high strength bolts and piezoelectric actuators that vary the contact force in the

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Piezoelectric actuator A 1

2

5 Friction interface

3

A

4

SMA wires

SectionA-A

Fig. 1. Schematic illustration and 3D rendering of hybrid device for Configuration I.

Piezoelectric actuator

A

SMA wires A

Friction interface

Section A-A

Fig. 2. Schematic illustration and 3D rendering of hybrid device for Configuration II.

friction interface based on controller command voltage. Superelastic SMA wires are attached to the ends of the inner and outer pipes. One of the advantages of this configuration is its straightforward and easy implementation. It does not require an external cover for protection of the wires, since SMA wires are located inside the pipes. Also, more SMA wire groups can be employed in the configuration I and the control of the contact force is easier in that configuration.

3. Modeling of re-centering variable friction device Fig. 3 shows the typical force–deformation curves of the sub-components of the RVFD, i.e. the SMA wires and friction device and the combined hysteresis. This section describes the models used to simulate the behavior of each subcomponent of the hybrid device proposed in this study.

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Force

Force

Force

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Displacement

Displacement Displacement

SMA

Combined

Friction Device Fig. 3. Force–deformation curves of the RVFD and its sub-components.

Conduct experimental tests to collect data

Generate generic FIS for ANFIS training

Validate the new model by using the validation data set

TRAIN FIS with ANFIS

Obtain optimized FIS from ANFIS

Fig. 4. Flow chart of fuzzy modeling of superelastic SMAs.

3.1. Modeling of SMA wires A neuro-fuzzy model is used to characterize the behavior of the superelastic SMA wires employed in the RVFD. A fuzzy inference system is a simple scheme that maps an input space to an output space using fuzzy logic. There are four main components of a FIS. They are: (i) a fuzzifier, which transforms the crisp inputs to fuzzy variables by defining membership functions to each input, (ii) a rule base, which relates the inputs to the output by means of if–then rules, (iii) an inference engine, which evaluates the rules to produce the system output and (iv) a defuzzifier, which transforms the output to a non-fuzzy discrete value. Among the various fuzzy models, the Seguno-type FIS [25] has attracted the most attention. The Seguno-type fuzzy models enable a systematic approach to model the dynamics of complex nonlinear systems. Adaptive neuro-fuzzy inference system (ANFIS) is a soft computing approach that combines fuzzy theory and neural networks [26]. In particular, the ANFIS employs neural network strategies to develop a Sugeno-type fuzzy model whose parameters (membership functions and rules) cannot be predetermined by user’s knowledge. One of the main advantages of the ANFIS is that it does not require a complex mathematical model to compute the system output. The ANFIS uses a hybrid algorithm to learn from the sample data from the system and can adapt the parameters inside its network. Here, the ANFIS is used to create a model of superelastic SMAs considering temperature and loading rate effects. Fig. 4 shows a flow chart for fuzzy modeling approach. First, an initial FIS is created which employs strain, strain rate and temperature as input variables and predicts the stress as single output. Next, data collected from experimental tests of NiTi wires performed at various loading rates and temperatures are concatenated in order to set up training, checking and validation data sets for the ANFIS simulations. After training, the developed FIS is validated using the data set reserved for validation and not used during the ANFIS training. Details in regard to the neurofuzzy model of the NiTi wires can be found in Ozbulut and Hurlebaus [21]. 3.2. Modeling of friction device A continuous hysteretic model proposed by Constantinou et al. [27] is used to represent the rigid-plastic behavior of the frictional force of the hybrid device as follows: Ff ¼ mNðtÞZ,

ð1Þ

where m represents the coefficient of friction; N is the contact force; and Z is a hysteretic dimensionless quantity expressed as n1

_ uy Z_ þ g9u99Z99Z9

n

_ þ bu9Z9 Au_ ¼ 0,

ð2Þ

where uy is the yield displacement; u_ is the slip velocity of the damper; and g, b, A and n are dimensionless parameters that control the shape of the hysteresis loop. The recommended values of these parameters to provide typical Coulomb friction

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behavior are uy ¼ 0.1 mm, g ¼0.5, b ¼0.5, A¼1 and n ¼2. Note that the hysteretic quantity Z is bounded by 71 and accounts for occurrence of the stick phase of the damper.

4. GA-based optimal fuzzy controller for RVFD The performance of a semi-active device highly depends on the control strategy that is used to adjust the normal force of the damper. Due to the uncertainties in the nature of ground motions and nonlinearities in structural system, as well as highly nonlinear force–displacement characteristics of the RVFD, development of a semi-active control law for modulation the voltage level of the RVFD using classical control strategies is challenging. One of the effective methods to deal with complex nonlinear systems is fuzzy logic approach. Fuzzy logic enables the description of relationships between inputs and outputs of a controller using simple verbose statements instead of complicated mathematical terms. In this study, a GA-based optimal fuzzy controller is developed to adjust the contact force at the friction interface by modulating the command voltage to the RVFD. In particular, a non-dominated sorting genetic algorithm, NSGA-II, is employed to design a self-organizing fuzzy controller. The NSGA-II algorithm creates an optimal set of solutions from an initial random population of solutions through mutation and recombination. First, it randomly generates an initial population and uniformly generates a set of individuals. After initialization, the NSGA-II uses a non-dominated sorting approach to rank individuals. A solution s1 is said to be dominated by the other solution s2 if the following conditions are true: (i) the solution s1 is no worse than the solution s2 in all objectives, (ii) the solution s1 is strictly better than the solution s2 in at least one objective. NSGA-II compares each solution with every other solution in the population to find if it is dominated, and the set of non-dominated individuals are assigned to the first front. The individuals in the second front are dominated only by those in the first front and the front goes so on. To provide diversity in the population, crowding distance, which is a measure of how close an individual is to its neighbor is calculated. Once the individuals are sorted by non-domination and with crowding distance assigned, parents are selected among the individuals that have (i) the highest rank or (ii) the highest crowding distance if they have the same rank. The offsprings are generated through crossover and mutation operators from the selected population. Finally, the NSGA-II yields a set of Pareto optimal solutions, among which one can select a solution that satisfies different goals to some extent. The detailed information on the algorithm can be found in Deb et al. [28]. In order to develop a fuzzy controller for the RVFD, first, inputs and output of a fuzzy logic controller (FLC) are defined and fuzzified into linguistic values. Then, the NSGA-II algorithm is used to tune the selected parameters and establish reasonable fuzzy rules. Here, the inputs of the fuzzy controller are selected as absolute accelerations of two different floors of a multi-story building and the output is the command voltage of the RVFD. Gaussian membership functions are used for two inputs and the single output. A Gaussian membership function can be defined using two parameters as follows: ðxcÞ2 mgauss ¼ exp  2s2

! ,

ð3Þ

where c is the location of the center and s is the width of the function. A typical rule in the fuzzy controller has the form: Rule i: IF X1 is Ai1 and X2 is Ai2 then Y is Ci, i¼1, 2, y, M,where M is the number of fuzzy rules, Xj are the input variables, Y is the output variable, and Aij and Ci are fuzzy sets characterized by membership functions mAij ðXj Þ and mCi ðYÞ, respectively. Each rule maps two input variables to a single output by relating corresponding membership functions. Therefore, in order to describe a single rule, three Gaussian membership functions are defined by using two parameters for each membership function. A total of 10 rules are generated to form the rule base by defining 10 membership functions for each of two inputs and single output. Consequently, 60 parameters are encoded into the GA chromosome to define the inference system of the FLC. To evaluate multi-objective fitness of each chromosome, four objective functions are defined and calculated as follows:    uj,cont:   , J1 ¼ max max  u j j,unc:

   uj,cont:   , J3 ¼ max rms uj,unc:  j

   u€ j,cont:   J2 ¼ max max u€ j,unc:  j    u€ j,cont:   J4 ¼ max rms u€ j,unc:  j

ð4Þ

where u and u€ denote interstory displacement and absolute story acceleration, respectively, and j represents the story that is considered. For the controlled case the RVFD are assumed to be present in the building. The performance indices J1 and J2 are based on the peak relative displacement and absolute acceleration of each floor, while the evaluation criteria J3 and J4 consider the entire duration of the motion and compute the root-mean-square (RMS) of the peak relative displacement and absolute acceleration for each floor. The objective functions from the response of the structure subjected to the Northridge earthquake described below are measured simultaneously and organized into a set of Pareto fronts.

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5. Numerical study In this section, nonlinear time history analyses of a three-story building are conducted to evaluate the efficacy of the RVFD in mitigating the seismic response of structures against near-field earthquakes. The properties of the building used in the simulations are given as follows [29]: the mass, initial elastic stiffness and viscous damping coefficient for each floor are mi ¼1000 kg, kei ¼980 kN/m and ci ¼1.407 kN s/m. The post-yielding stiffness of each floor is aikei, where ai ¼0.1 for i¼ 1, 2 and 3, and the yielding levels for each floor are uy1 ¼22.6 mm, uy2 ¼16.8 mm and uy3 ¼11.2 mm. The RVFD is rigidly connected between the ground and first floor as shown in Fig. 5. A nonlinear block that uses a Bouc–Wen model to relate the deformation history and restoring force with nonlinear characteristics is developed in MATLAB and Simulink [30] to perform the simulations. The equation of motion for a three

m3

..

u3

k3

m2

k2 m1

..

u1 Voltage

RVFD

k1 Fuzzy Controller

Fig. 5. Three-story building with RVFD.

0.5

1940 El Centro

Acc. (g)

Acc. (g)

0.5 0 -0.5

-0.5 0

10

0.5

20

0

30

10

20

0.5

1995 Kobe

Acc. (g)

Acc. (g)

1994 Northridge

0

0

30

1999 Chi-Chi

0 -0.5

-0.5 0

10

20 Time (s)

30

0

20

40 60 Time (s)

Fig. 6. Time histories of the selected ground motions.

80

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degree-of-freedom system is expressed as mi ðu€ i þ u€ g Þ þ FSi þFRVFD di1 ¼ ð1di3 ÞFSði þ 1Þ ,

ð5Þ

where mi is the mass of each floor; u€ g is the ground acceleration, ui and FSi are the relative displacement and restoring force of each floor, respectively; FRVFD is the lateral force exerted by the RVFD ; and dij is the Kronecker delta which is one if its variables i and j are equal, and zero otherwise. Restoring force FSi is defined as follows: FSi ¼ ci u_ i þ kei ui þ ð1ai Þkei uyi zi ,

ð6Þ

where zi is given as z_ i ¼

o   u_ i n n Ai  bi þ gi sgnðu_ i zi Þ 9zi 9 , uyi

ð7Þ

Uncontrolled

Controlled

3

3

2

2

2

1

1

1

El Centro

Floor

3

0

20

40

60

0

5

10

15

0

3

3

2

2

2

1

1

1

20

30

Northridge

Floor

3

10

0

20

40

60

0

5

10

15

0 3

2

2

2

1

1

1

10

20

30

Kobe

Floor

3

3

0

20

40

60

0

3

3

2

2

5

10

15

0

10

20

30

3

Floor

Floor

Floor

Chi-Chi

1

1

0

20

40

Peak interstory drift (mm)

60

2

1

0

5

10

Peak absolute acceleration

15 (m/s2)

0

10

20

30

Residual displacement (mm)

Fig. 7. Profiles of peak story drift, peak absolute acceleration, and residual displacement for uncontrolled and controlled cases under various excitations.

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where Ai, bi, gi, and n are shape parameters for hysteresis loops, and have the values of 1, 0.5, 0.5, and 1, respectively. Note that in Eq. (6) the first term represents the damping force due to inherent damping and the sum of the second and third term is the nonlinear restoring force of the structure. The absolute accelerations of the first and third floors are selected as inputs to the fuzzy controller. After a trial and error procedure, the area and length of the SMA wires used for the RVFD are selected to be 0.18 cm2 and 50 cm for the considered numerical example. Also, in the hybrid device, the capacity of the friction device mNmax is chosen to be 6.3 kN with a preload of Npre ¼3 kN. The response of the structure is evaluated under the fault-normal components of the 1940 El Centro earthquake, 1994 Northridge earthquake, 1995 Kobe earthquake, and 1999 Chi-Chi earthquake. All ground motion records are scaled to a peak ground acceleration of 0.5g. Acceleration time histories of the selected ground motions are given in Fig. 6. Nonlinear time history analyses of the three story building are performed in order to evaluate the performance of the RVFD. 5.1. Structural response In this section, first, the profiles of peak response quantities for all excitation cases are summarized, and then displacement and acceleration time histories as well as hysteresis loop of shear force of each floor and force–deformation curve of the RVFD are presented for Kobe earthquake. Fig. 7 shows the profiles of peak relative displacement, residual displacement and peak absolute acceleration that result from the numerical simulations for the three-story building with and without control. It can be seen that the relative

Uncontrolled 1 1st

40

1st Floor

Floor Acceleration (m/s2)

Interstory drift (mm)

60

20 0 -20 -40 5

10

15

20

25

0 -0.5

30

0

60

5

10

15

20

25

30

1.5 2nd Floor

40

Acceleration (m/s2)

Interstory drift (mm)

0.5

-1 0

20 0 -20 -40

2nd Floor

1 0.5 0 -0.5 -1 -1.5

0

5

10

15

20

25

30

0

15

5

10

15

20

25

30

1.5 3rd Floor

10

Acceleration (m/s2)

Interstory drift (mm)

Controlled

5 0 -5 -10 -15

3rd Floor

1 0.5 0 -0.5 -1 -1.5

0

5

10

15 Time (s)

20

25

30

0

5

10

15 Time (s)

20

25

30

Fig. 8. Relative displacement and absolute acceleration time histories of the each floor for uncontrolled and controlled systems.

Force (kN)

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20

20

20

10

10

10

0

0

0

-10

-10 1st Floor

-20

Force (kN)

-20

0

20

-10 2nd Floor

-20 -20

40

0

20

3rd Floor -20

40

-20

20

20

20

10

10

10

0

0

0

-10

-10 1st Floor

-20

0

20

40

-10 3rd Floor

2nd Floor -20

-20 0 20 40 Interstory drift (mm)

2857

-20 -20 0 20 40 Interstory drift (mm)

-20 0 20 40 Interstory drift (mm)

Fig. 9. Hysteresis loops of inelastic shear force for each floor with and without control.

displacements of the structure are significantly reduced when the RVFD is installed into the structure. In particular, the reductions in the peak interstory drift are 42%, 41%, 50%, and 39% for the El Centro, Northridge, Kobe and Chi-Chi earthquakes, respectively. The maximum acceleration responses are also reduced by 6–14%, except for the Chi-Chi earthquake, which is increased only by 1%. In addition to peak displacement and acceleration responses of the structure which are important parameters to define the performance of the structures, residual deformations are important in evaluating the post-earthquake vulnerability of the structure. It can be seen that the uncontrolled building experiences large permanent drifts after the earthquake whereas there is no significant residual deformations for the controlled structure for most of the cases. Specifically, excessive residual deformations that are observed in the first floor of the structure under all ground motions considered here are prevented with the application of the RVFD. However, in the case of the Chi-Chi earthquake, considerable residual deformations are present in the second floor of the controlled structure. This result suggests that using the RVFD at multiple floors can be more effective in preventing residual deformations when the structure is subjected to severe earthquakes. Fig. 8 compares the time histories of relative displacement and absolute acceleration at each floor of the uncontrolled and controlled systems for the Kobe earthquake. The effectiveness of the RVFD in improving the displacement response of the structure is seen from the figure. Note that especially interstory drifts of the first and second floors which are substantial for the uncontrolled case significantly reduced with the application of the RVFD. Although the amplitude of the maximum acceleration response at each floor is similar for controlled and uncontrolled cases, the acceleration response over the entire duration of the seismic event is smaller for the controlled building. Fig. 9 shows the hysteresis loops of inelastic shear force for each floor. Note that without control large yield deformations are observed at the first and second floor, while for the controlled case yielding do not occurs for the first floor and less yielding occurs for the second floor.

5.2. Energy consideration Earthquake-induced vibrations of structures can be described as an energy transferring process. When an earthquake occurs, energy is produced as a result of the movement of a fault and released in the form of seismic waves. As a result, certain amounts of energy are input to the structure from the ground motion, which in return induces structural vibrations. This input energy is transformed to other energy forms such as kinetic energy and strain energy. Therefore, studying the energy characteristics and energy transfer can be helpful in making decision on implementing innovative seismic protection systems [31].

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The absolute energy balance equation of an N-storey building subjected to an earthquake ground motion can be written as [32] Z t Z t Z t X  1 T u_ t Mu_ t þ mj u€ tðjÞ dug FTS du ¼ ð8Þ u_ T C du þ 2 0 0 0 where ut is the vector of absolute displacements of the system, u is the vector of relative displacements of the system, _ ¼ ðN  1Þ vector of restoring forces, mj is the M¼(N  N) diagonal mass matrix, C¼ (N  N) damping matrix, and FS ½u, u lumped mass of the jth story and ug is the ground displacement. Eq. (8) can be simply expressed as EK þEx þ EA ¼ EI ,

ð9Þ

where EK ¼

Ex ¼

EA ¼

1 T u_ t Mu_ t 2 Z

t

u_ T C du ¼

Z

0

Z Z 0

t

u_ T Cu_ dt

ð11Þ

0 t

0

EI ¼

ð10Þ

t

FTS du ¼ X

Z

t

u_ T FS dt

ð12Þ

0

Z  mj u€ tðjÞ dug ¼

t

X

0

 mj u€ tðjÞ u_ g dt

ð13Þ

Here, EK denotes the absolute kinetic energy, Ex is the damping energy, EA is the absorbed energy, which is composed of the recoverable elastic strain energy ES and the irrecoverable hysteretic energy EH, and EI represents the absolute input energy. For the controlled structure, the left-hand side of Eq. (9) has an additional term ED, which is the energy dissipated Uncontrolled 5 Damping Hysteretic Input

Energy (kN-m)

4 3 2 1

0 0

5

10

15 Time (s)

20

25

30

20

25

30

Controlled 5 Damping Hysteretic RVFD Input

Energy (kN-m)

4 3 2 1

0 0

5

10

15 Time (s)

Fig. 10. Energy time histories for the uncontrolled and controlled structures subjected to Kobe earthquake.

O.E. Ozbulut, S. Hurlebaus / Mechanical Systems and Signal Processing 25 (2011) 2849–2862

by the RVFD and this term can be computed as Z t u_ T FRVFD dt ED ¼

2859

ð14Þ

0

where FRVFD is the vector of damper force. Fig. 10 shows the energy distribution time history of different forms for the uncontrolled and controlled structure subjected to Kobe earthquake. It can be seen that most of the input energy is dissipated by inelastic deformations for the uncontrolled structure. On the other hand, the hysteretic energy is considerably lower for the controlled structure, indicating that less damage takes place for a structure with the RVFD and the energy dissipation is mostly provided by the RVFD. It should be also noted that the total energy input to the structure decreases when the RVFD is installed. To further evaluate the effectiveness of the RVFD, the ratio of the energy dissipated by the RVFD to the energy dissipated by the structure is computed as follows:

Z¼

ED Ex þEH

ð15Þ

which is illustrated in Fig. 11. Note that for a given time a value larger than one for Z indicates that the RVFD device is dissipating more energy than the structure. It can be seen that the RVFD is effectively dissipating the input energy over the entire duration of the ground motion. Specifically, Z goes up to as much as 5 and then attains a value of less than 2, which indicates the RVFD dissipates more energy than the structure itself for all times.

6 5



4 3 2 1 0 0

5

10

15 Time (s)

20

25

30

Fig. 11. Effectiveness of RVFD as a function of time.

2 SMA VFD Energy (kN-m)

1.5

1

0.5

0 0

5

10

15 Time (s)

20

25

Fig. 12. Energy time histories for the sub-components of the RVFD.

30

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The time histories of the energy dissipated by the sub-component of the RVFD are shown in Fig. 12. It can be seen that the energy is dissipated mainly through friction and the SMA components of the RVFD have small contribution to the energy dissipation. This is expected since the SMA wires are used in the hybrid device to provide mainly re-centering capability. Several researchers have questioned the ability of the total energy concept in differentiating the difference between damaging effects of impulsive near-field earthquake and cyclic far-field earthquake [33]. Hori and Inoue [34] proposed the momentary input energy as an index for representing damage potential of an earthquake to a structure. Momentary input energy describes the increase of energy during unit time. Amiri et al. [35] found out that the maximum momentary input energy is more appropriate parameter than the maximum total input energy for estimating the damage potential of ground motions with near-field characteristics to multi-story buildings. Therefore, in the following, the momentary input energy is also computed to investigate the performance of the proposed hybrid device in reducing the response of structures against near-field earthquakes. The momentary input energy is defined as

DE ¼

Z

t þ Dt

mT u€ t dug ¼

t

Z

t þ Dt

mT u€ t u_ g dt

ð16Þ

t

where DE represents the absolute momentary input energy for time step Dt input to a structure. For a single-degree-offreedom system, Dt is mostly chosen to be the half-cycle period of the hysteresis loop, which is the time step between the

ΔE / ΔT (kN-m/s)

Uncontrolled 2 0 -2 0

ΔE / ΔT (kN-m/s)

NORTHRIDGE

4

5

10

15 Time (s)

20

25

Controlled

0 -2 0

5

10

15 Time (s)

20

25

-2

Controlled

2 0 -2 0

5

10

15

0 -20

6 4 2 0 -2

40

Uncontrolled

45

5

10

15

20

Time (s)

Time (s) 6

40

ΔE / ΔT (kN-m/s)

ΔE / ΔT (kN-m/s)

15

KOBE

Uncontrolled

35

10

Time (s)

40

30

5

4

30

ΔE / ΔT (kN-m/s)

ΔE / ΔT (kN-m/s)

0

0

CHI-CHI

20

Uncontrolled

2

Time (s)

4 2

4

30

ΔE / ΔT (kN-m/s)

ΔE / ΔT (kN-m/s)

EL CENTRO

Controlled

20 0 -20

4

Controlled

2 0 -2

30

35

40 Time (s)

45

5

10

15 Time (s)

Fig. 13. Time histories of momentary input energy of uncontrolled and controlled structure subjected to different earthquakes.

20

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maximum positive and negative displacement responses. With that choice of Dt, the computed DE corresponds to the maximum momentary input energy DEmax. Since for a multi-degrees-of-freedom system, the peak displacement response of different stories does not necessarily occur simultaneously, the choice of Dt to obtain DEmax is more difficult. In this study, Eq. (16) is computed for Dt¼ 0.02, 0.04, y, T/2, where T is the fundamental period of the structure and, the Dt corresponding to the maximum momentary energy is chosen as desired time step. Fig. 13 illustrates the time histories of momentary input energy divided by Dt for the uncontrolled and controlled structure subjected to the ground motions described above. It can be seen the momentary input energy is considerably reduced when the structure is controlled with the RVFD. For example, there is a 49% decrease in the momentary input energy for the structure subjected to Chi-Chi earthquake when the RVFD is installed. It can be seen from Fig. 7 that the peak displacement response of the structure subjected to the Chi-Chi earthquake experiences a similar decrease for the controlled case. However, the maximum total input energies for the uncontrolled and controlled structure subjected to Chi-Chi earthquake are very close to each other and differ by only 9%. This suggests that the use of momentary input energy as an index for evaluating the damage potential of near-field ground motions is more appropriate than the use of the total input energy. 6. Conclusions This study proposes a re-centering variable friction device for seismic protection of structures subjected to near-field earthquakes. In the RVFD, which exhibits a double flag-shaped hysteresis loop, SMA wires are used to achieve re-centering ability while its energy dissipation capacity is improved through friction. Specifically, a piezoelectric stack driven by a controllable voltage is used to regulate the normal force of the hybrid damper in order to provide the desirable level of energy dissipation. Further, a fuzzy controller developed by employing a multi-objective genetic algorithm determines the voltage to send to the piezoelectric stack. A neuro-fuzzy model is used to describe the behavior of the SMA components of the hybrid device while a continuous hysteretic model is used to characterize the friction force developed. Nonlinear time-history analyses of a three-story building with an installed hybrid device between the ground and the first floor are conducted under various near-field earthquakes. Peak structural response quantities of each floor such as peak interstory drift, peak absolute acceleration and residual deformation are evaluated. 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