Control of Structural Response Under Earthquake ... - Semantic Scholar

Computer-Aided Civil and Infrastructure Engineering 27 (2012) 620–638

Control of Structural Response Under Earthquake Excitation Unal Aldemir,∗ Arcan Yanik & Mehmet Bakioglu Faculty of Civil Engineering, Division of Mechanics, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey

Abstract: This study firstly proposes some representative simple methods to obtain the suboptimal passive damping and stiffness parameters from the optimal control gain matrix since it is not possible to add the exact optimal damping and stiffness parameters to the structure in practice. It is shown numerically that modifying the structural damping and the stiffness in the proposed suboptimal ways may suppress the uncontrolled vibrations while the performance levels depend on the seismic inputs. Since the proposed approach is intrinsically passive and has no adaptive property against changing dynamic effects, this study secondly proposes a new performance index so that the mechanical energy of the structure, control and the seismic energies are considered simultaneously in the minimization procedure. The implementation of the resulting closedloop control algorithm does not require both a priori knowledge of the seismic excitation and the solution of the nonlinear matrix Riccati equation. The performance of the proposed approach is investigated, e.g., structures subjected to three seismic inputs and compared to the performance of the uncontrolled, the classical linear optimal control, and the passive cases. It is shown by the numerical simulation results that the proposed algorithm is capable of suppressing the uncontrolled seismic structural displacements and the absolute accelerations simultaneously and performs almost as well as the classical linear optimal control in reducing the displacements with comparable control effort and performs better than the classical linear optimal control in reducing the absolute accelerations. The results show that while the proposed active approach has similar performance to the classical linear optimal control, the classical linear optimal control increases the absolute accelerations slightly compared to the proposed active approach in regulating displacements, while the proposed active approach regulates ∗ To

whom correspondence should be addressed. E-mail: aldemiru@ itu.edu.tr.

 C 2012 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/j.1467-8667.2012.00776.x

and reduces both displacements and absolute accelerations. The proposed approach is promising in protecting both the structural and non-structural members from the seismic forces since a simultaneous reduction both in the displacements and the absolute accelerations is achieved. 1 INTRODUCTION Conventional civil engineering structures are designed relying on the mass and rigidity of the structure to resist to the uncertain dynamic loads. The need of adaptability to resist uncertain loads increased the safety levels. The strong desire for better utilization of new materials and lower costs motivated the development of new concepts for protecting structures. The new alternative approaches such as passive, semiactive, and active control have been proposed and developed to protect structures from earthquakes and severe winds (Yao, 1972; Soong, 1990; Adeli and Saleh, 1997; Saleh and Adeli, 1998; Naeim and Kelly, 1999; Lin et al., 2010; Ou et al., 2010; Bitaraf et al., 2012). To improve the structural performance there has been an increasing progress in research and development of these innovations (Saleh and Adeli, 1994; Gluck et al., 1996; Sing and Moreschi, 2001; Adeli and Kim, 2004; Kim and Adeli, 2005a; Miranda, 2005; Reinhorn et al., 2009; Chey et al., 2010; Bitaraf et al., 2011; Cho et al., 2012). Among the passive devices, base isolation systems are mostly researched and widely applied in practice (Riley et al., 1998; Agrawal et al., 2006; Narasimhan et al., 2006). Base isolation systems are effective in reducing the inter-story displacements. However, the excessive overturning moments for the base isolated multi-story buildings and the excessive base displacements due to the near fault excitations are some of the issues to be addressed for these systems (Palazzo and Petti, 1999).

Control of earthquake response of structures

Another commonly used passive device is the tuned mass damper (Wang et al., 2001; Aldemir, 2003; Sadek et al., 1997; Kang et al., 2012). Although they are widely studied, the effectiveness of these devices is limited due to the mistuning effect (Chey et al., 2010). If the tuning frequency of the mass damper differs from the main frequency of the structure, tuned mass damper will have little effect in reducing the seismic responses. Casciati and Guilliani (2009) reported about weak seismic performance of the tuned mass dampers with very small mass ratios. With the developments of digital control and sensor techniques, the research in the semi-active and active control techniques has increased (Bakioglu and Aldemir, 2001; Aldemir and Bakioglu, 2001; Aldemir, 2009; Adeli and Jiang, 2009; Aldemir and Gavin, 2006; Alhan et al., 2006; Gavin and Aldemir, 2005). Active control methods are effective for a wide frequency range as well as for the transient vibrations. Since the 1990s, active control systems have also found increasing applications to mitigate the seismic hazards in civil engineering structures. Some of these control methods include, for instance, optimal control (Aldemir et al., 2001; Alavinasab et al., 2006), adaptive control (Diamantis et al., 2011; Theodoridis et al., 2010), frequency domain techniques (Suhardjo et al., 1992), fuzzy control (Nomura et al., 2007; Kim et al., 2010, Wu et al., 2010), H∞ control (Schmitendorf et al., 1994; Wang, 2011), and decentralized control algorithms (Lei et al., 2012). Most of the studies in the literature are based on the classical active control algorithms, which are the applications of the regulator problem in which the performance index is defined as the integration of a quadratic expression with respect to the state and the control vectors. This performance index has two contributions; the first one reflects the desire of bringing the controlled variable to zero while the second one that of keeping the control input as small as possible. But, only classical closed-loop control can be applied to structures. However, since the nonlinear matrix Riccati equation is obtained by ignoring the earthquake excitation term classical closed-loop control is approximately optimal and does not satisfy the optimality condition. On the other hand, while the classical closed-open loop control and open loop algorithms are superior to the closed-loop control, they are not applicable to earthquake-excited structures, because the whole earthquake ground acceleration history is not known a priori. Therefore, it is almost impossible to find the optimal control exactly for the structures under earthquake forces. In research studies and practical applications, various control algorithms have been investigated to overcome this deficiency of the classical control algorithms (Yang

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et al., 1987; Sato and Toki, 1990; Xu et al., 2005; Akhiev et al., 2002; Aldemir, 2010). The work of Adeli and Kim (2004, 2009) who developed a novel wavelet-hybrid feedback-Linear Mean Square (LMS) algorithm for robust control of large civil structures is an innovative work in this area. The authors, in fact, are widely recognized for introducing the concept of wavelets (Adeli and Samant, 2000; Jiang and Adeli, 2005; Lin et al., 2012) in structural control for the first time. Their work has also two distinct contributions. It is applied to vibration control of large and realistic building (Kim and Adeli, 2005b, c) and bridge structures (Kim and Adeli, 2005a). Their algorithm is especially effective for vibration control of irregular high-rise building structures which are prone to damage during severe earthquake loading (Kim and Adeli, 2005b). Kim and Adeli presented a hybrid control system through judicious combination of a passive supplementary damping system with a semi-active tuned liquid column damper system (Kim and Adeli, 2005d). Jiang and Adeli (2008a) presented a new nonlinear control model for active control of high-rise threedimensional building structures considering both geometrical and material nonlinearities as well the coupling between lateral and torsional motions of the structure and coupling between the actuator and the structure. The model is based through ingenious integration of wavelets, a signal processing technique (Zhou and Adeli, 2003), chaos theory, and two soft computing techniques, neural networks (Puscasu and Codres, 2011), and fuzzy logic. They apply the model to two irregular three-dimensional steel building structures, a 12-story structure with vertical setbacks, and an 8-story structure with plan irregularity. Jiang and Adeli (2008b) present a floating-point genetic algorithm (Mathakari et al., 2007; Sarma and Adeli, 2001) for finding the optimal control forces needed for active nonlinear control of high-rise building structures. This study firstly proposes a simple method to determine the additional passive stiffness and damping parameters via the closed-loop classical linear optimal control gain matrix. Since the exact optimal stiffness and damping matrices cannot be added in general directly to the stiffness and damping matrices of the structure, the corresponding suboptimal story damping and stiffness parameters are determined approximately in some simple representative ways. Since the additional passive parameters are determined suboptimally and the proposed system is a passive system that has no adaptive character against dynamic loadings, the passive structure with the suboptimal damping and stiffness suppresses the uncontrolled vibrations while its performance varies for different seismic inputs. The proposed study secondly introduces the seismic energy

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term into the performance index so that the mechanical energy of the structure, control and the seismic energies are considered simultaneously in the minimization procedure, which yields cross terms in the performance index. Even though the proposed performance index has the disturbance effect through the seismic energy, the implementation of the resulting closed-loop control algorithm does not require both a priori knowledge of earthquake and the solution of the nonlinear matrix Riccati equation. The effectiveness of the derived control is investigated by applying the algorithm to example linear structures incorporating active tendon system under three earthquakes. Numerical simulation results indicate that the control policy derived from the proposed index achieves almost the same reduction in displacements as classical linear optimal control by comparable control energy associated with the smaller absolute accelerations than those of the classical linear optimal control. It is known that the earthquake engineers usually deal with the dilemma of balancing displacement and absolute accelerations. Even if the structural relative acceleration is eliminated, the structure would still experience the accelerations due to the control forces and the seismic ground motion. So, it is desirable to reduce the absolute accelerations accounting for both of these additional contributions. Achieving a simultaneous reduction both in the displacements and the absolute accelerations indicates that the proposed approach is promising in protecting both the structural and nonstructural members from the seismic forces.

in which X(0) = X(1) = 0 in practice. Upon introduc˙ T , this system ing the (2n × 1) state vector Z = (X, X) given by Equation (1) is easily converted to a linear, first-order state-space system defined as: ˙ Z(t) = A Z(t)+B U(t)+D f (t), t ∈ (t0 , t1 ) in which



0 −M−1 K   0 D= −η

A=

   0 I ;B = ; −M−1 C M−1 D2

(3)

(4)

and I is an (n × n) identity matrix, the η = (1, . . . ,1)T is an (n × 1) constant vector. Initial condition of the system Equation (3) can be written using Equation (2) as follows;  (0)  X 0 (5) Z(t0 ) = Z = X(1) Solution of Equation (3) with Equation (5) can be derived in the following form; t Z(t) = e

A(t−t0 )

Z + 0

eA(t−s) q(s) ds

(6)

t0

where q(t) =B U(t)+D f (t)

(7)

The following equation is easily obtained from Equation (6): t

2 CLASSICAL LINEAR OPTIMAL CONTROL

Z(t) =e

At

Z(t − t) +

eA(t−s) q(s) ds

(8)

t−t

A shear-beam lumped mass linear building under the one-dimensional earthquake excitation and the control can be idealized by an n-degree of freedom system. Equations of motion of the structure may be expressed as ¨ ˙ M X(t) + C X(t) + K X(t) = D1 f (t) + D2 U(t), t ∈ (t0 , t1 )

(1)

where X(t) = (X 1 , X 2, . . . , Xn) T is the (nx1) relative displacement vector; M, C, and K are the (nxn) mass, damping, and stiffness matrices, respectively; D1 = -(m1 , . . . , mn) T is the (nx1) location matrix of the excitation; D2 is the (nxr) location matrix of r controllers; U(t) = (u1 (t), . . . , ur (t))T is the (rx1) control force vector and the scalar function f (t) is the one-dimensional ground acceleration. Initial conditions of the structure can be given as ˙ 0 ) = X(1) X(t0 ) = X(0) ; X(t

(2)

where t is the sampling interval. After evaluating the integral on the right-hand side of Equation (8) with the aid of trapezoidal rule, Z(t) can be expressed as Z(t) = eAt Z(t − t) + (t/2)eAt q(t − t) + (t/2)[B U(t) + D f (t)] + O(t 3 )

(9)

where O(t 3 ) denotes the quantity g(t) which satisfies the condition |g(t)| ≤ C0 t 3 ,C0 = constant > 0 for t → +0. In the classical optimal control law; the classical integral type quadratic performance measure t1 (ZT QC Z + UT RC U) dt

J=

(10)

0

is minimized where t1 is the duration longer than that of earthquake. Minimization yields the following linear optimal control U(t) = −

1 −1 T R B P(t) Z(t) 2 C

(11)

Control of earthquake response of structures

In Equation (11), P is the symmetric, positive-definite solution to 1 −1 T ˙ P(t) + P(t)A − P(t)B R−1 C B RC B P(t) (12) 2 + AT P(t) + 2 QC = 0 ; P(t1 ) = 0 where the subscript C refers to classical linear optimal control; QC and RC are user defined positive semidefinite and positive definite weighting matrices. It is highly desirable to develop systematic methods to select the weighting matrices, which will guarantee the stability of the controlled structure. Unfortunately, there are no systematic approaches or exact rules in general on how to assign the values of the weighting matrices. The tuning of the weighting matrices reflects the trade-off between the desired performance and the control energy consumption, and it provides the designer with significant flexibility. So, determining the optimal values of the weighting matrices is an issue of this trade-off. However, within the framework of the present study, these weighting matrices are required to guarantee the stability of the controlled structure. One can assign the numerical values to the elements of the weighting matrices and then check the stability condition by computing the eigenvalues of the controlled system, which will involve trial and error procedures. Equation (11) can also be expressed as U(t) = GC Z(t)

(13)

where the control gain matrix GC (n×2n) is 1 GC (t) = − R−1 BT P(t) (14) 2 C The Riccati matrix P(t) obtained from Equation (12) does not yield an optimal solution since the excitation term f (t) vanishes within the control interval [0, t1 ] (Soong, 1990), or it yields a solution which corresponds to white noise disturbance (Sage and White, 1977; Kwakernaak and Sivan, 1972). It is also known that the P(t) can be assumed to be almost constant in practice for structural control applications. Therefore, it is concluded from Equations (12) and (14) that once the numerical values are assigned to the elements of the weighting matrices for a given structure, then the corresponding Riccati matrix and the gain matrix are constant and independent from the earthquake excitations.

3 PROPOSED CONTROL APPROACHES This study firstly investigates the effect of adding suboptimal stiffness and damping on the structural response and then proposes a new performance index for the active control of structures.

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3.1 Passive approach Using the sub-matrices G1 and G2 , the control gain GC given by Equation (14) can be expressed as GC = [G1

G2 ]

(15)

Substituting Equation (15) into Equation (13) yields   X(t) U = [G1 G2 ] ˙ (16) X(t) where G1 and G2 are (n×n) sub-matrices of GC (Gluck et al., 1996; Sing and Moreschi, 2001; Reinhorn, 2009). Substituting Equation (16) into Equation (1) yields the following equation of the motion of the structure ˙ ¨ + [K − D2 M X(t) + [C − D2 G2 ]X(t) = D1 f (t)

G1 ]X(t) (17)

It is clear from Equation (17) that the control force modifies the damping and stiffness matrices of the uncontrolled structure. This modification can be expressed as Kopt = − D2 G1 ; Copt = − D2 G2

(18)

In most cases, these optimal stiffness and damping matrices obtained from the classical linear optimal control law cannot be directly added to the damping and stiffness matrices of the structure. It seems not possible to extract the additional story damping and stiffness parameters which are equivalent to the exact optimal stiffness and damping matrices. If this is possible, then there will be no need for the implementation of active devices. However, since this is not possible in general, this study proposes some simple representative approaches to determine the additional story damping and stiffness parameters which are not exact optimal, but sub-optimal. If the exact optimal stiffness and damping matrices have an appropriate form, they are added directly to the stiffness and damping matrices of the structure. If not, the eigenvalues of the optimal stiffness and damping matrices may be added to the structure as representative values of the optimal stiffness and damping matrices. Since the eigenvalues can be considered to be the representative values of the optimal stiffness and damping matrices, the corresponding reasonable amount of sub-optimal damping and stiffness can be added in practice to the structures in various ways such as increasing the dimensions of load carrying members, implementing viscous dampers and bracings. However, the corresponding representative damping may be too large compared to actual values in practice for some actuator positions and the selected parameters. Investigating the optimal distribution of the excessive damping along the structure is a different research topic which is beyond the scope of this article. Some very simple

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representative approaches are defined below for comparison purposes. The eigenvalues of the optimal stiffness and damping matrices may be added to the corresponding stiffness and damping coefficients of the structure as follows: Eigi (−D2 G2 ) = Cai ; i = 1, n Eigi (−D2 G1 ) = Kai ; i = 1, n

(19)

which is the seismic energy input to the structure and D1 = -(m1 , . . . , mn) T . J2 can be obtained from the energy equilibrium equation of the structure as follows: t ˙ (τ )M X(τ ¨ )dτ + X

= Ka j ; j = 1, n Or, the maximum damping and stiffness parameters can be assigned to the first floor while the minimum damping and stiffness parameters can be assigned to the top floor as given below: [Eigi (−D2 G2 )]max,...,min = Ca1,...,n ; [Eigi (−D2 G1 )]max,...,min = Ka1,...,n

0

0

Finally, assuming the exact optimal stiffness and damping matrices as symmetric band matrices similar to the classical structural stiffness and damping matrices with identical elements, the corresponding identical floor damping and stiffness parameters can be obtained approximately as 1  1  Kopt (i, i) = Ka j ; Copt (i, i) 2n − 1 2n − 1 (22) i=1

i=1

= Ca j ; j = 1, n

0

+

J = J1 (Z,U) + J2 (Z,t)

(23)

˙ T (τ )D2 U(τ )dτ X

The first term on the left-hand side of Equation (25) is the kinetic energy of the system which is ˙ T (t)M X(t); ˙ (1/2)X the second term is the damping energy and the third term is the elastic strain energy which is (1/2)XT (t)K X(t). The first term of the right-hand side of Equation (25) is the seismic energy input to the structure which is denoted as J2 and the second term is the control energy. J2 can be obtained from Equation (25) as t J2 =

˙ TM X ˙ TC X ˙ T (τ ) D1 f (τ )dτ = 1 X ˙ + t X ˙ X 2 2 (26) 0 t ˙ T t T X D2 U + E(t − t) + X KX− 2 2

where E(t-t) contains the energy terms within the time interval between 0 and t-t. J2 can also be expressed in terms of the state and control force vectors as J2 = ZT Q 2 Z + ZT H1 U + E(t−t)

J1 = Z (t)Q1 Z(t) + U (t) R U(t) t ˙ T (τ ) D1 f (τ )dτ J2 = X

0 0 t t M K C+ 2 2 2



 ;

H1 =

(27)

 0 t (28) − D2 2

The proposed performance function can now be given as J = ZT Q 1 Z + UT R U + [ZT Q 2 Z + ZT H 1 U (29) + E(t−t)] J = ZT (Q 1 + Q 2 )Z + UT R U + ZT H 1 U (30) + E(t−t)

where T

0

0

Q2 = This study proposes the following functional so that the mechanical energy of the structure, control and the seismic energies are considered simultaneously in the minimization procedure,

˙ T (τ )D1 f (τ )dτ (25) X

t



3.2 Active control approach

t ˙ T (τ )K X(τ )dτ = X

+

(21)

n

˙ T (τ )C X(τ ˙ )dτ X

t

where Cai and Kai denote the additional damping and the stiffness parameters for the ith story. Or, average damping and stiffness parameters can be assigned to stories as follows: n n 1 1 Eigi (−D2 G2 ) = Ca j ; Eigi (−D2 G1 ) n n (20) i=1 i=1

n

t T

T

(24)

0

In Equation (24), J1 is the time dependent quadratic scalar functional; Q1 is the positive semi-definite weighting matrix; J 2 is an integral type quadratic functional

It is noted here that the proposed index has a cross term of state and control vectors. The necessary conditions of optimality for the proposed performance function to be minimum under the constraints given by Equations (2) and (3) are obtained by Lagrange multiplier method using the Hamiltonian of

Control of earthquake response of structures

the problem,    t H = J + λ Z(t) − eAt Z(t − t) + q(t − t) 2 t − q(t) (31) 2 in which λ is the Lagrangian multiplier vector (Chung et al., 1995; Lin et al., 1996; Singh and Moreschi, 2001). Then, the necessary conditions of optimality are as follows: ∂H ∂H ∂H = 0; = 0; =0 (32) ∂λ ∂U ∂Z From Equations (31) and (32), the following equations are obtained,  t t q(t − t) − q(t) = 0 Z(t) − eAt Z(t − t) + 2 2 (33) 2 R U + HT1 Z −

t T T B λ =0 2

2(Q 1 + Q 2 )Z +H1 U + λT = 0

(34) (35)

Equation (35) can also be written as 2Q Z + H1 U + λT = 0

(36)

where Q = Q1 + Q2 . If Equation (34) is rewritten using Equation (36), the following equation is obtained   t T t T 2R+ B H1 U + B 2 Q Z + HT1 Z = 0 (37) 2 2 Then the optimal control force is obtained from the above equation as U(t) = −Gpc Z(t)

(38)

where the gain matrix Gpc is −1 

 t T B H1 Gpc = 2 R + t BT Q + HT1 (39) 2 In Equation (39), the subscript “pc” refers to the proposed control. Substituting Equation (38) into Equation (3) yields the following equation of the optimally controlled structure Z˙ = [A − B Gpc ]Z + D f (t)

(40)

  ˙ = A − B(2 R + t BT H1 )−1 (t BT Q + HT1 ) Z Z 2 (41) + D f (t) Equation (41) can also be rewritten as ˙ = N Z + D f (t) Z where



t T N = [A − B 2R + B H1 2

(42)

−1 (t BT Q + HT1 ) (43)

625

If the real parts of the eigenvalues of the closed loop system matrix N in Equation (43) lie in the left half of the complex plane, then the system is asymptotically stable. Implementation of the derived control force does not require the solution of nonlinear Riccati matrix equation. Since it is not a closed-open loop control, it does not depend on the disturbance and need the future values of the ground motion.

4 NUMERICAL EXAMPLE To investigate the effectiveness of the proposed control algorithm and the passive approach, a 3- and a 10-story shear-beam type structure with tendon controllers are analysed. These structures and the investigated cases are shown in Figures 1a–d. Some important issues in practice such as modeling the structure in three-dimensional complex form, measurement noise, computational time delays, phase lag effects, and the determination of the optimal placement of the controllers are beyond the scope of this study and have not been taken into account in derivations for the sake of simplicity. Since the focus of this study is on examining the performance limitations of the simple passive approach in protecting the earthquake excited structures and illustrating the improvement associated with the proposed new performance index, even if the selected 3and 10-story shear-type structures are simple compared to more realistic models, they may still be accepted to be sufficient for comparison purposes of the proposed algorithms. These structures are analysed under three earthquake loadings with different properties to be able to reach the general conclusions; the Erzincan (1992), Northridge (1994), and simulated ground motion. The acceleration time histories of these earthquakes are shown in Figure 2. Simulated ground motion is generated by the formula ¨ X¨ 0 (t) = m(t) X(t)

(44)

where m(t) is the deterministic nonnegative envelope ¨ is a stationary random process with zero function, X(t) mean and a power spectral density function S(ω). As spectral density function, the following well known Kanai-Tajimi power spectral density function is used S(ω) =

1 + 4ξg2 (ω/ωg )2 [1 − (ω/ωg

2 )2 ]

+

4ξg2 (ω/ωg )2

S2

(45)

Here, ωg , ξg , and S are the characteristic frequency of the soil, predominant damping coefficient, and the intensity of the soil motion, respectively. These parameters are selected as 18.85 rd/sec, 0.65 and 0.00465 m2 /sec3 /rd, respectively. The deterministic nonnegative

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Aldemir, Yanik & Bakioglu

Fig. 1. Schematics of 3- and 10-story structure models equipped with active tendons.

compare the performance of the proposed algorithms and the results are given in tables for all the example structures and control cases. The performance measures are defined for the first story, except β3 which is defined for the total seismic and control energy of the structure since the most critical responses for the structural safety are observed in general for that story. However, the maximum response quantities of all floors are given in tables. The investigated performance measures in terms of the maximum values are defined as   max |X1 | CLOC max  X¨ 1  PC   , , β2 = β1 = max |X1 |NC max  X¨ 1 NC   (47)  X¨ 1abs  CLOC max |CE| max PC   β3 = , β4 = max |SE| max  X¨ 1abs  NC

Fig. 2. Earthquake acceleration time histories: (a) Erzincan, (b) Northridge, and (c) synthetic.

where subscript “NC” refers to uncontrolled structure, “PC” refers to proposed control, and “CLOC” refers to classical linear optimal control.

envelope function is given as:

4.1 Example structure I

m(t) = 0, t < 0 ; m(t) = (t/t1 )2 , 0 ≤ t ≤ t1 m(t) = 1, t1 ≤ t ≤ t2 ; m(t) = e−c(t−t2 ) , t2 ≤ t

(46)

where t1 , t2 , and c are the parameters that should be selected appropriately to reflect the shape and duration of the earthquake ground acceleration (Yang et al., 1987). The parameters associated with the ground acceleration model are selected as t1 = 3 sec, t2 = 13 sec, and c = 0.26 sec−1 . Simulated ground motion has a maximum acceleration of 0.33 g. Four performance measures which take into account the displacement, the relative acceleration, the absolute acceleration, the seismic energy (SE), and the control energy (CE) of the structure are defined to be able to

For the investigated 3-story structure; each story has a mass of 100 tons, an elastic stiffness coefficient of 15,791 kN/m, and a linear viscous damping coefficient of 125.66 kNsec/m. For the 3-story structure with single tendon controller, the weighting matrix RC (1×1) for the classical linear optimal control (CLOC) is chosen to be 0.50 × 10−4 ; R(1×1) is chosen to be 0.3 x10−4 for the proposed control (PC). For the full-tendon controller case, the weighting matrix RC (3×3) for the CLOC is chosen to be 0.5 × 10−4 I3×3 ; R(3×3) is chosen to be 0.4 x10−4 I3×3 for PC. The weighting matrices QC and Q1 are partitioned as follows:     K C 03×3 03×3 QC = ; Q1 = 100 (48) C M C M

Control of earthquake response of structures

627

The weighting matrices are selected as to achieve approximately same level control forces for both PC and CLOC. Since the real parts of the all eigenvalues of the closed-loop system matrix N lie in the left half of the complex plane (−22.3312, −22.3312, −10.6949, −10.6949, −1.3623, −1.3623), the controlled system is asymptotically stable. The control gain matrix GC for the structure with a single tendon controller is calculated as [987.4; −1,263.5; 544.6; 193.1; −418.7; −363.2] and then the corresponding Kopt and Copt are obtained as ⎡ ⎤ 987.4 −1,263.5 544.6 Kopt = ⎣ −987.4 1,263.5 −544.6 ⎦ 0 0 0 ⎡ ⎤ (49) 193.1 −418.7 −363.2 363.2 ⎦ Copt = ⎣ −193.1 418.7 0 0 0

rectly added to the stiffness matrix of the structure while the eigenvalues of Copt are added to the damping coefficients of the stories. In case C4, the eigenvalues of these two matrices are added to the stiffness and damping coefficients of the corresponding stories. The eigenvalues of these matrices are calculated as Eig(Kopt ) = (3,150.2; 1,508.6; 192.2) and Eig(Copt ) = (255.6, 611.4, 794.5). It must be noted here that the damping values obtained from the exact optimal damping matrix for structure I with the selected actuator positions and the weighting parameters are too large compared to usual values in practice. Investigating the optimal distribution of the excessive damping along the structure is beyond the scope of this article. The above-defined very simple representative approaches are given for comparison purposes.

These matrices are different in arrangement from the conventional structural stiffness and damping matrices. Since it is not possible to add these matrices directly to the stiffness and damping matrices of the structure, as a possible simple approach, the corresponding eigenvalues may be added. The eigenvalues of these matrices are: Eig(Kopt ) = (0; 2,251; 0) and Eig(Copt ) = (0, 611.8, 0). For the first passive approach case (C1), the second eigenvalue of stiffness matrix has been added directly to the stiffness coefficient of the first story and the second eigenvalue of damping matrix has been added directly to the damping coefficient of the first story. In the second case (C2), the eigenvalues of the stiffness and damping matrices are simply equally distributed to the floors. For the 3-story structure with a full-tendon controller system; GC , Kopt , and Copt are calculated as

4.2 Example structure II

GC = ⎡ 970.20 ⎣ −970.20 −0.02

−0.03 970.20 −970.20

0.03 437.83 0.01 −199.22 970.2 −50.52

238.61 387.31 −199.22

⎤ 188.09 238.61 ⎦ 437.83

(50) ⎡

Kopt

Copt

⎤ 1,940.40 −970.2 0.0 = ⎣ −970.20 1,940.40 −970.20 ⎦ 0.0 −970.20 970.20 ⎡ ⎤ 637.05 −148.70 −50.52 = ⎣ −148.70 586.53 −199.22 ⎦ −50.52 −199.22 437.83

(51)

Equation (51) shows that Kopt is similar to the conventional stiffness matrix in arrangement for k1 = k2 = k3 = 970.2 kN/m, while Copt cannot be directly added to the damping matrix since Copt does not have a conventional damping matrix form. In case C3, Kopt is di-

A 10-story structure with 3 and 10 tendon controllers is examined (Figures c and d). The mass, stiffness, and damping coefficients of each story are selected as m = 345.6 tons, k = 3.404 × 105 kN/m, and c = 2,937 tons/sec. The actuators are implemented at the first, fifth, and the tenth floors for 3-tendon case while each floor has 1 tendon controller for the 10-tendon case. More information about the optimal placement of active controllers in structures can be found in the literature (Pantelides and Cheng, 1990; Abdullah et al., 2001). The weighting matrices QC and Q1 are partitioned as follows (Loh et al., 1999; Wong and Yang, 2001a, b):     K 010×10 K 010×10 ; Q1 = q2 QC = q1 010×10 M 010×10 M (52) In Equation (52), appropriate values are assigned to the parameters q1 and q2 to be able to achieve approximately same level control forces for both PC and CLOC. For the three-tendon case, the weighting matrix RC (3×3) is chosen to be 0.5 × 10−2 I3×3 for CLOC; R(3×3) is chosen to be diagonal matrix with diagonal elements of 0.15 × 10−4 I3×3 for PC. For the 10-tendon case, RC (10×10) is chosen to be 0.2 × 10−3 I10×10 for CLOC and R(10×10) is chosen to be 0.5×10−5 I10×10 for PC. For the three-tendon case, Kopt and Copt are given in Table 1. Table 1 indicates that Kopt and Copt are different from the conventional stiffness and damping matrix and it is not possible to obtain the equivalent exact optimal story stiffness and damping parameters. Eigenvalues of these two matrices are obtained as Eig(Kopt ) = (56,858; 22,912; 1,204.8; 0; 0; 0; 0; 0; 0; 0) and Eig(Copt ) = (4,808.2; 3,169.2; 618.22; 0; 0; 0; 0; 0; 0; 0). These

628

Aldemir, Yanik & Bakioglu

Table 1 Kopt and Copt for 10-story structure with three tendon controllers Kopt =

15,500.0 −5,340.0 0.0 15,500.0 −5,340.0 0.0 0.0 0.0 −5,340.0 −1,770.0

−5,620.0 15,300.0 0.0 −5,620.0 15,300.0 0.0 0.0 0.0 15,300.0 −7,710.0

−1,300.0 −618.0 0.0 −1,300.0 −618.0 0.0 0.0 0.0 −618.0 1,190.0

11,500.0 −4,850.0 0.0 11,500.0 −4,850.0 0.0 0.0 0.0 −4,850.0 −764.0

−9,200.0 17,200.0 0.0 −9,200.0 17,200.0 0.0 0.0 0.0 17,200.0 −7,980.0

−1,680.0 −2,320.0 0.0 −1,680.0 −2,320.0 0.0 0.0 0.0 −2,320.0 2,750.0

−147.0 −3,890.0 0.0 −147.0 −3,890.0 0.0 0.0 0.0 −3,890.0 2,860.0

1,610.0 −2,430.0 0.0 1,610.0 −2,430.0 0.0 0.0 0.0 −2,430.0 1,060.0

−4,130.0 13,700.0 0.0 −4,130.0 13,700.0 0.0 0.0 0.0 13,700.0 −8,010.0

−503.0 −9,330.0 0.0 −503.0 −9,330.0 0.0 0.0 0.0 −9,330.0 7,740.0

Copt =

1,290.0 69.0 0.0 1,290.0 69.0 0.0 0.0 0.0 69.0 −456.0 0.0

279.0 1,440.0 0.0 279.0 1,440.0 0.0 0.0 0.0 1,440.0 −1,020.0 0.0

683.0 795.0 0.0 683.0 795.0 0.0 0.0 0.0 795.0 −717.0 0.0

1,360.0 563.0 0.0 1,360.0 563.0 0.0 0.0 0.0 563.0 −786.0 0.0

−147.0 1,910.0 0.0 −147.0 1,910.0 0.0 0.0 0.0 1,910.0 −1,180.0 0.0

−324.0 1,250.0 0.0 −324.0 1,250.0 0.0 0.0 0.0 1,250.0 −677.0 0.0

−488.0 1,080.0 0.0 −488.0 1,080.0 0.0 0.0 0.0 1,080.0 −470.0 0.0

−797.0 1,310.0 0.0 −797.0 1,310.0 0.0 0.0 0.0 1,310.0 −423.0 0.0

−1,290.0 1,890.0 0.0 −1,290.0 1,890.0 0.0 0.0 0.0 1,890.0 −457.0 0.0

−967.0 446.0 0.0 −967.0 446.0 0.0 0.0 0.0 446.0 704.0 0.0

eigenvalues are directly added to the stiffness and damping coefficients of the structure for the case C5. For the case C6, the maximum eigenvalues of these two matrices are added to the stiffness and damping coefficient of the first story and the other eigenvalues are added to the stories in a decreasing way to the upper stories (for three-tendon case, C5 and C6 cases are the same). In the case C7, if the off-diagonal terms are small enough compared to the diagonal elements, then the optimal damping and stiffness matrix can be treated approximately as a conventional structural damping and stiffness matrix in which the stories are identical. In this case, the sum of the diagonal elements of the optimal matrices is divided by 19 and the resulting value is added equally to the stiffness and damping coefficients of the stories. For the last case C8, an average damping and stiffness value is obtained by dividing the sum of the eigenvalues of the Kopt and Copt by 10 and then these average values are added to the stiffness and damping coefficients of the stories. For the 10-tendon controller case, Kopt and Copt are given in Table 2. The eigenvalues of these two matrices are calculated as Eig(Kopt ) = (76,041; 71,022; 63,158; 53,092; 41,799; 30,223; 19,439; 434.82; 3,860.4; 10,384) and Eig(Copt ) = (717.42; 1,815.4; 2,557; 3,053.5; 3,385.6; 3,609.7; 3,759.9; 3,860.8; 3,958.9; 3,923.9). It must be noted here that the damping values obtained from the exact optimal damping matrix are comparable to the existing story damping values for both 3 and 10 actua-

tor cases. For the case C5, the eigenvalues of these two matrices are directly added to the stiffness and damping matrices of the stories. Table 2 indicates that the offdiagonal elements of Kopt are very small and the offdiagonal elements of Copt are small compared to the diagonal elements and it can be modified so that it will have a similar form to conventional stiffness matrix. For this reason, in the case C7, the sum of the diagonal elements of the optimal stiffness and damping matrix are divided by 19 and the equivalent story stiffness and damping parameters are found as k = 19,450 kN/m and c = 1,612.74 tons/sec/m, respectively. For the case C8, the sum of the eigenvalues of these two matrices are divided by 10 and added to the stiffness and damping coefficients of the stories. The maximum response quantities and control forces of 3-story structure with single and three tendon controllers are given in Tables 3 and 4 for all the investigated cases. The maximum response quantities and control forces of 10-story structure with three tendon controllers under Erzincan, Northridge, and synthetic earthquakes are given in Tables 5–7 for all the investigated cases, respectively. The maximum response quantities and control forces of 10-story structure with 10 tendon controllers under Erzincan, Northridge, and synthetic earthquakes are given in Tables 8–13 for all the investigated cases, respectively. The performance measures for the 3-story structure with single and 3 tendon controllers are given in Tables 14 and 15 for all the

Control of earthquake response of structures

629

Table 2 Kopt and Copt for 10-story structure with 10 tendon controllers Kopt =

38,880.0 −19,443.0 −10.7 16.6 −4.8 7.9 −3.6 −2.9 1.1 −1.9

−19,455.0 38,889.0 −19,467.0 34.8 −15.8 18.9 −4.7 −7.1 2.9 −4.4

−1.2 −19,452.0 38,866.0 −19,403.0 −30.0 33.4 −4.3 −11.6 8.7 −10.5

−3.8 −3.8 −19,448.0 38,898.0 −19,461.0 36.5 −9.2 −5.5 16.7 −21.7

−10.2 13.0 −12.8 −19,441.0 38,889.0 −19,429.0 −19.5 34.0 1.0 −25.9

−1.3 7.9 −8.6 4.9 −19,447.0 38,891.0 −19,447.0 16.5 18.7 −32.1

−1.2 12.6 −11.4 4.7 1.8 −19,454.0 38,898.0 −19,434.0 0.9 −13.0

3.6 8.0 −8.6 4.5 3.4 −8.4 −19,443.0 38,896.0 −19,451.0 1.2

4.5 3.7 −2.3 2.6 1.7 −6.4 2.1 −19,453.0 38,890.0 −19,437.0

−4.2 9.4 0.2 −1.3 7.6 −12.9 2.3 −9.4 −19,446.0 19,456.0

Copt =

3,399.1 −470.6 −201.6 −97.5 −53.2 −32.1 −21.5 −15.9 −12.6 −11.6

−470.6 3,197.4 −568.5 −254.2 −129.9 −74.4 −47.9 −34.6 −26.9 −24.6

−201.4 −568.8 3,144.0 −599.7 −276.4 −145.0 −87.3 −60.0 −45.3 −40.5

−97.5 −255.1 −600.5 3,122.6 −616.2 −288.0 −156.8 −99.4 −71.6 −62.4

−53.0 −129.7 −276.6 −616.5 3,110.3 −627.5 −300.3 −169.0 −114.3 −95.1

−31.7 −74.6 −145.9 −289.1 −627.7 3,098.7 −640.0 −315.9 −190.6 −148.3

−20.9 −47.6 −87.7 −157.1 −300.4 −640.7 3,083.4 −661.9 −348.1 −245.2

−15.0 −33.8 −59.6 −98.8 −169.8 −316.5 −661.8 3,050.7 −714.9 −446.5

−12.0 −26.9 −45.7 −72.0 −114.6 −191.5 −348.3 −715.6 2,953.4 −916.8

−11.0 −23.7 −39.9 −61.3 −93.4 −147.2 −244.5 −446.6 −916.4 2,482.4

Table 3 Maximum response quantities for 3-story structure with single tendon controller Earthquake Erzincan

Northridge

Synthetic

max Xi (m) NC

max Xi (m) C1

max Xi (m) C2

max Xi (m) PC

max ur (kN) PC

max Xi (m) CLOC

max ur (kN) CLOC

0.1589 0.2811 0.3438 0.0893 0.1617 0.2008 0.0426 0.0712 0.0851

0.1223 0.2333 0.2916 0.0538 0.0994 0.1261 0.0270 0.0477 0.0636

0.1393 0.2469 0.3043 0.0653 0.1150 0.1428 0.0312 0.0530 0.0677

0.1413 0.2327 0.2889 0.0578 0.0958 0.1196 0.0275 0.0404 0.0546

– – 592.26 – – 293.64 – – 218.15

0.1312 0.2274 0.2792 0.0560 0.0953 0.1174 0.0258 0.0387 0.0498

– – 597.24 – – 318.51 – – 215.79

Table 4 Maximum response quantities for 3-story structure with three tendon controllers Earthquake Erzincan

Northridge

Synthetic

max Xi (m) NC

max Xi (m) C3

max Xi (m) C4

max Xi (m) PC

max ur (kN) PC

max Xi (m) CLOC

max ur (kN) CLOC

0.1589 0.2811 0.3438 0.0893 0.1617 0.2008 0.0426 0.0712 0.0851

0.127 0.222 0.2724 0.0512 0.0900 0.1112 0.0197 0.0338 0.0406

0.1136 0.2064 0.2595 0.0414 0.0768 0.0977 0.0164 0.0304 0.0379

0.0901 0.1583 0.1947 0.0375 0.0660 0.0814 0.0151 0.0264 0.0323

542.97 420.53 235.23 284.97 188.21 99.96 221.61 163.4 93.83

0.0906 0.1597 0.1962 0.0358 0.0635 0.0787 0.0142 0.0251 0.0312

513.08 425.43 246.39 273.28 194.78 105.78 226.59 180.60 99.32

630

Aldemir, Yanik & Bakioglu

Table 5 Maximum response quantities for 10-story structure with three tendon controllers. Input: Erzincan Displacement Xi (i = 1, . . . ,10) (m) Story no. NC PC CLOC C5 C6 C7 C8

Control force (kN)

1

2

3

4

5

6

7

8

9

10

0.0696 0.0212 0.0226 0.0584 0.0584 0.0686 0.0677

0.1373 0.0415 0.0429 0.1207 0.1207 0.1352 0.1334

0.2007 0.0612 0.0634 0.1833 0.1833 0.1976 0.1951

0.2582 0.0784 0.0813 0.2405 0.2405 0.2543 0.2511

0.3088 0.0933 0.0955 0.2910 0.2910 0.3042 0.3005

0.3520 0.1079 0.1113 0.3342 0.3342 0.3470 0.3429

0.3877 0.1204 0.1237 0.3698 0.3698 0.3823 0.3778

0.4153 0.1295 0.1318 0.3974 0.3974 0.4095 0.4047

0.4342 0.1355 0.1360 0.4163 0.4163 0.4281 0.4230

0.4438 0.1412 0.1415 0.4259 0.4259 0.4375 0.4323

1

5

10

1635.6 1645.5

3193.2 3242.2

2283.2 1778.1

Table 6 Maximum response quantities for 10-story structure with three tendon controllers. Input: Northridge Displacement Xi (i = 1, . . . ,10) (m) Story no. NC PC CLOC C5 C6 C7 C8

Control force (kN)

1

2

3

4

5

6

7

8

9

10

0.0531 0.0071 0.0079 0.0412 0.0412 0.0512 0.0495

0.1048 0.0134 0.0135 0.0848 0.0848 0.1008 0.0973

0.1538 0.0209 0.0223 0.1288 0.1288 0.1476 0.1424

0.1987 0.0290 0.0303 0.1692 0.1692 0.1905 0.1835

0.2385 0.0352 0.0364 0.2049 0.2049 0.2284 0.2199

0.2723 0.0426 0.0437 0.2354 0.2354 0.2607 0.2508

0.2996 0.0487 0.0496 0.2602 0.2602 0.2867 0.2759

0.3201 0.0532 0.0538 0.2790 0.2790 0.3063 0.2949

0.3336 0.0559 0.0562 0.2916 0.2916 0.3194 0.3076

0.3403 0.0586 0.0584 0.2979 0.2979 0.3259 0.3139

1

5

10

871.57 794.27

1365.4 1461.4

956.63 851.54

Table 7 Maximum response quantities for 10-story structure with three tendon controllers. Input: Synthetic Displacement Xi (i = 1, . . . ,10) (m) Story no. NC PC CLOC C5 C6 C7 C8

Control force (kN)

1

2

3

4

5

6

7

8

9

10

0.0149 0.0065 0.0052 0.0104 0.0104 0.0130 0.0138

0.0288 0.0121 0.0092 0.0211 0.0211 0.0248 0.0265

0.0413 0.0176 0.0137 0.0320 0.0320 0.0358 0.0380

0.0523 0.0222 0.0172 0.0420 0.0420 0.0465 0.0487

0.0622 0.0255 0.0193 0.0511 0.0511 0.0562 0.0590

0.0720 0.0284 0.0223 0.0591 0.0591 0.0647 0.0681

0.0806 0.0306 0.0260 0.0659 0.0659 0.0719 0.0758

0.0875 0.0339 0.0289 0.0713 0.0713 0.0776 0.0820

0.0923 0.0364 0.0310 0.0750 0.0750 0.0816 0.0863

0.0948 0.0383 0.0328 0.0769 0.0769 0.0836 0.0885

1

5

10

494.7 562.3

825.6 1004.3

482.3 593.1

Table 8 Maximum displacement responses for 10-story structure with 10 tendon controllers. Input: Erzincan Displacement Xi (i = 1, . . . ,10) (m) Story no. NC PC CLOC C5 C6 C7 C8

1

2

3

4

5

6

7

8

9

10

0.0196 0.0116 0.0100 0.0217 0.0203 0.0209 0.0218

0.0390 0.0224 0.0195 0.0429 0.0403 0.0414 0.0430

0.0571 0.0322 0.0283 0.0636 0.0598 0.0606 0.0630

0.0730 0.0410 0.0362 0.0835 0.0786 0.0780 0.0814

0.0865 0.0488 0.0435 0.1026 0.0967 0.0932 0.0979

0.0977 0.0554 0.0500 0.1205 0.1136 0.1061 0.1124

0.1071 0.0607 0.0554 0.1364 0.1288 0.1167 0.1246

0.1146 0.0648 0.0596 0.1499 0.1413 0.1250 0.1343

0.1198 0.0675 0.0626 0.1593 0.1502 0.1308 0.1409

0.1224 0.0689 0.0641 0.1640 0.1549 0.1338 0.1444

Control of earthquake response of structures

631

Table 9 Maximum control forces for 10-story structure with 10 tendon controllers. Input: Erzincan Control force (kN) Story no. PC CLOC

1

2

3

4

5

6

7

8

9

10

2,942.1 3,146.5

2,759.1 3,041.8

2,543.9 2,869.5

2,327.1 2,632.7

2,108.2 2,342.5

1,868.2 2,014.9

1,587.6 1,652.1

1,254.7 1,266.5

871.19 858.76

446.6 435.4

Table 10 Maximum displacement responses for 10-story structure with 10 tendon controllers. Input: Northridge Displacement Xi (i = 1, . . . ,10) (m) Story no. NC PC CLOC C5 C6 C7 C8

1

2

3

4

5

6

7

8

9

10

0.0531 0.0233 0.0208 0.0422 0.0411 0.0490 0.0451

0.1048 0.0455 0.0406 0.0829 0.0809 0.0963 0.0885

0.1538 0.0663 0.0593 0.1214 0.1186 0.1408 0.1291

0.1987 0.0854 0.0767 0.1571 0.1535 0.1815 0.1661

0.2385 0.1024 0.0924 0.1893 0.1851 0.2175 0.1989

0.2723 0.1170 0.1059 0.2177 0.2128 0.2481 0.2269

0.2996 0.1290 0.1167 0.2415 0.2361 0.2729 0.2499

0.3201 0.1382 0.1248 0.2604 0.2542 0.2917 0.2674

0.3336 0.1444 0.1302 0.2733 0.2665 0.3043 0.2793

0.3403 0.1475 0.1329 0.2798 0.2728 0.3106 0.2852

Table 11 Maximum control forces for 10-story structure with 10 tendon controllers. Input: Northridge Control force (kN) Story no. PC CLOC

1

2

3

4

5

6

7

8

9

10

2,107.9 2,152.6

2,035.6 2,110.3

1,938.1 2,020

1,805.6 1,884

1,639.2 1,701.6

1,435.6 1,474.7

1,197.4 1,213.1

928.38 928.06

633.86 625.64

321.61 315.65

Table 12 Maximum displacement responses for 10-story structure with 10 tendon controllers. Input: Synthetic Displacement Xi (i = 1, . . . ,10) (m) Story no. NC PC CLOC C5 C6 C7 C8

1

2

3

4

5

6

7

8

9

10

0.0149 0.0066 0.0063 0.0109 0.0103 0.0105 0.0102

0.0288 0.0125 0.0119 0.0211 0.0199 0.0202 0.0195

0.0413 0.0178 0.0168 0.0307 0.0289 0.0290 0.0276

0.0523 0.0224 0.0210 0.0397 0.0372 0.0370 0.0347

0.0622 0.0263 0.0245 0.0480 0.0448 0.0447 0.0409

0.0720 0.0296 0.0276 0.0555 0.0517 0.0519 0.0470

0.0806 0.0322 0.0304 0.0620 0.0579 0.0583 0.0521

0.0875 0.0341 0.0327 0.0676 0.0631 0.0635 0.0560

0.0923 0.0355 0.0342 0.0713 0.0668 0.0672 0.0587

0.0948 0.0362 0.0350 0.0731 0.0689 0.0691 0.0601

Table 13 Maximum control forces for 10-story structure with 10 tendon controllers. Input: Synthetic Control force (kN) Story no. PC CLOC

1

2

3

4

5

6

2,376 1,849

1,895.8 1,754.3

1,626.8 1,663.1

1,471.2 1,549

1,328 1,388.3

1,164.2 1,205.7

7 975.17 1,005.4

8

9

10

759.31 795.42

520.71 553.58

264.91 288.45

632

Aldemir, Yanik & Bakioglu

Table 16 Performance measures for 10-story structure with three tendon controllers

Table 14 Performance measures for 3-story structure with single tendon controller Earthquake Erzincan

Northridge

Synthetic

Control cases

β1

β2

β3

β4

NC C1 C2 CLOC PC NC C1 C2 CLOC PC NC C1 C2 CLOC PC

1.00 0.77 0.88 0.83 0.89 1.00 0.60 0.73 0.63 0.65 1.00 0.63 0.73 0.65 0.61

1.00 0.75 0.81 0.94 0.87 1.00 0.81 0.89 1.02 0.78 1.00 0.84 0.94 1.08 1.05

– – – 0.62 0.57 – – – 0.73 0.72 – – – 0.71 0.73

1.00 0.85 0.89 0.87 0.91 1.00 0.82 0.81 0.90 0.75 1.00 0.81 0.78 0.91 0.66

Earthquake Erzincan

Northridge

Synthetic

Table 15 Performance measures for 3-story structure with three tendon controllers Earthquake Erzincan

Northridge

Synthetic

Control cases

β1

β2

β3

β4

NC C3 C4 CLOC PC NC C3 C4 CLOC PC NC C3 C4 CLOC PC

1.00 0.80 0.71 0.57 0.57 1.00 0.57 0.46 0.40 0.42 1.00 0.46 0.38 0.33 0.35

1.00 0.79 0.77 0.69 0.59 1.00 0.77 0.76 0.77 0.60 1.00 0.92 0.90 0.85 0.79

– – – 0.91 0.92 – – – 0.90 0.91 – – – 0.93 0.94

1.00 0.89 0.90 0.85 0.75 1.00 0.66 0.62 0.62 0.54 1.00 0.53 0.61 0.69 0.57

investigated cases, respectively, while the performance measures for the 10-story structure with 3 and 10 tendon controllers are given in Tables 16 and 17, respectively, for all the investigated cases. As evident from Tables 3 and 4, the passive C1, C2, C3, and C4 suppress the uncontrolled response significantly. However, it must be noted here that the added suboptimal damping values obtained from the exact optimal damping matrix are extremely higher than those of the existing structure. This indicates also that the exact optimal performance cannot be achieved by the suboptimal passive damping in practice. These cases are included for comparison purposes. It is also clear from

Control cases

β1

β2

β3

β4

NC C5 C6 C7 C8 CLOC PC NC C5 C6 C7 C8 CLOC PC NC C5 C6 C7 C8 CLOC PC

1.00 0.84 0.84 0.98 0.97 0.32 0.30 1.00 0.77 0.77 0.96 0.93 0.15 0.13 1.00 0.70 0.70 0.87 0.93 0.35 0.43

1.00 0.83 0.83 0.99 0.98 0.81 0.70 1.00 0.84 0.84 0.75 0.75 0.93 0.83 1.00 0.83 0.83 0.95 0.92 1.02 0.91

– – – – – 0.95 0.96 – – – – – 0.91 0.92 – – – – – 0.88 0.89

1.00 0.99 0.99 0.99 0.99 1.14 1.11 1.00 0.92 0.92 0.98 0.96 1.05 0.90 1.00 0.98 0.98 0.99 0.99 1.15 1.06

Table 17 Performance measures for 10-story structure with 10 tendon controllers Earthquake Erzincan

Northridge

Synthetic

Control cases

β1

β2

β3

β4

NC C5 C6 C7 C8 CLOC PC NC C5 C6 C7 C8 CLOC PC NC C5 C6 C7 C8 CLOC PC

1.00 1.11 1.04 1.06 1.11 0.51 0.59 1.00 0.79 0.77 0.92 0.85 0.39 0.43 1.00 0.73 0.69 0.70 0.68 0.42 0.44

1.00 0.96 0.88 0.96 0.93 0.88 0.66 1.00 1.01 0.97 1.00 0.98 0.87 0.61 1.00 0.91 0.83 0.87 0.83 0.85 0.64

– – – – – 0.91 0.92 – – – – – 0.86 0.86 – – – – – 0.94 0.93

1.00 1.01 1.00 0.98 0.98 1.04 1.03 1.00 0.97 0.94 0.96 0.93 0.87 0.67 1.00 0.95 0.90 0.94 0.90 0.95 0.81

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633

Fig. 3. Absolute acceleration responses for the 3-story structure with single tendon controller.

Fig. 4. Absolute acceleration responses for the 3-story structure with three tendon controllers.

Fig. 5. Absolute acceleration responses for the 10-story structure with three tendon controllers.

Tables 3 and 4 that the performance levels of the passive C1, C2, C3, and C4 cases for the 3-story structure are almost comparable to each other in terms of the reduction in the uncontrolled response. The above given results can also be observed from Tables 14 and 15 in terms of the performance measures ß1 , ß2 , and ß4 . The performance measure ß4 is less than one for the passive C1, C2, C3, and C4 cases. It can also be concluded from Tables 3 and 4 and Tables 14 and 15 that the passive C1, C2, C3, and C4 cases show different performance in

response reduction for different earthquakes. This is a typical characteristic of passive systems since they have no ability to adapt themselves to dynamic loads. Tables 5–7 and Table 16 show that the passive C5 and C6 approaches are identical as noted before and perform better than the passive C7 and C8 cases, but perform worse than the passive C1, C2, C3, and C4 in terms of suppressing the uncontrolled response. It is clear from Table 16 that the performance measure ß4 is less than one for all the passive cases of C5, C6, C7, and C8.

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Fig. 6. Absolute acceleration responses for the 10-story structure with 10 tendon controllers.

Fig. 7. Displacement (X1 , X2 , X3 ) time histories for the 3-story structure with three tendon controllers. PC (thick solid), NC (thin solid): (a) Erzincan, (b) Northridge, and (c) synthetic.

It is also observed from Tables 5 to 7 and Table 16 that PC and CLOC outperform all the passive cases of C5, C6, C7, and C8. It is obtained from Tables 8, 10, 12, and 17 for the 10-story structure with 10 controllers that all the passive cases of C5, C6, C7, and C8 suppress the uncontrolled response for Northridge and synthetic earthquakes, but slightly increase the uncontrolled response for Erzincan earthquake. These results indicate that the optimal damping and stiffness parameters for a given structure may be different for different earthquakes. The optimal structural parameters for a specific earthquake may not be optimal for another ground motion and may increase the structural response much more than expected. In fact, this is one of the major reasons of considering semi-active or active systems to improve the seismic structural performance.

It is obvious from Tables 3 to 10 and Tables 14 to 17 that PC and CLOC algorithms achieve a considerable decrease in the uncontrolled response and outperforms all the investigated passive cases. Remembering that ß3 denotes the ratio of the total control energy to the total seismic energy input to the structure as defined above, it is observed from Tables 3 and 14 that both CLOC and PC may achieve the same level reduction in response with almost the same percentage of control energy for these cases. Table 14 shows also that PC yields slightly smaller or almost the same absolute acceleration for the first floor (ß4 ) than CLOC for all the investigated excitations. Tables 4 and 15 (3-story structure with three controllers) show that PC achieves almost the same reduction in the response with almost the same control effort (ß3 ) as CLOC for all the investigated excitations

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Fig. 8. Displacement (X1 , X5 , X10 ) time histories for the 10-story structure with 10 tendon controllers. PC (thick solid), NC (thin solid): (a) Erzincan, (b) Northridge, and (c) synthetic.

Fig. 9. Energy time histories for the 10-story structure with three tendon controllers. SE = seismic energy, CE = control energy: (a) Erzincan, (b) Northridge, and (c) synthetic.

while PC yields a smaller absolute acceleration for the first floor (ß4 ) than CLOC. The similar results are also observed for 10-story structure with 3 controllers (Tables 5–7 and Table 16) and for 10-story structure with 10 controllers (Tables 8–13 and 17) for all the investigated seismic excitations. The maximum floor absolute accelerations for the 3-story structure and the 10-story structure are shown in Figures 3–6. As shown in these figures, PC performs in general slightly better than CLOC in reducing the absolute accelerations for all the investigated cases. As noted in the introduction, the structure may still experience the accelerations due to the control and seismic forces even if the relative accelerations are mitigated. So, it is important to reduce the absolute accelerations accounting for both of these additional contributions. Achieving a simultaneous reduction both in the displacements and the absolute accelerations indicates that the proposed approach may protect both the structural and non-structural members from the seismic forces.

The uncontrolled displacement time histories of the 3-story structure with 3 controllers and the 10-story structure with 10 controllers are compared to those of PC in Figures 7 and 8 for all the investigated excitations. Figures 7 and 8 show that PC achieves a significant reduction in the uncontrolled response. As evident from Table 16, the ratio of the control energy to the total seismic input energy (ß3 ) corresponding to the investigated excitations is almost the same for PC and CLOC. The time histories of the control and seismic input energies for the 10-story structure with three controllers under the investigated seismic excitations are also shown in Figure 9 for illustrative purposes. As shown in Figures 9b and c for the Northridge and synthetic earthquakes, both the total seismic input energy and the energy ratio ß3 for PC and CLOC are very close to each other. However, Figure 9a for the Erzincan earthquake illustrates that the total seismic energy input depends on the applied control algorithm and are different for PC and CLOC while the corresponding

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energy ratio (ß3 ) is almost the same. It is clear from the data presented that PC has the potential of suppressing the uncontrolled response almost as well as CLOC by almost the same percentage of control effort and reducing the absolute accelerations more than CLOC.

regulates and reduces both displacements and absolute accelerations simultaneously to mitigate the structural damage and the occupant injury.

ACKNOWLEDGMENT 5 CONCLUSIONS This article firstly proposes some simple passive representative approaches to determine the floor suboptimal damping and stiffness to be added to the structure since the exact optimal stiffness and damping matrices cannot be added in general directly to the stiffness and damping matrices of the structure. It is shown numerically that the modifying of the damping and the stiffness in a suboptimal way may suppress the uncontrolled vibrations. However, it may also be possible for these passive parameters to increase the responses for some excitations since the additional passive parameters are not optimal and the proposed system is a passive system that has no adaptive character against dynamic loadings. To be able to overcome this deficiency of the passive systems, the present study proposes also a new performance index for the vibration suppression of earthquake-excited structures. The implementation of the resulting control scheme does not require the solution of the nonlinear matrix Riccati equation and a priori knowledge of the seismic excitation. Performance of the proposed algorithm is investigated through the numerical simulations of the example linear structures subjected to recorded and synthetic earthquakes and compared to those of the uncontrolled case, passive cases, and the classical linear optimal control. The results show that the proposed control outperforms the passive approaches for all the investigated cases and can significantly reduce both the displacements and the absolute accelerations of the uncontrolled structures. It is known that even if the floor relative acceleration is suppressed, the structure would still experience the accelerations due to the control forces and the seismic ground motion. So, it is desirable to limit the absolute accelerations accounting for both of these additional contributions. Numerical results illustrate that the resulting control policy of the proposed index achieves almost the same reduction in displacements as classical linear optimal control by comparable control energy associated with the smaller absolute accelerations than those of the classical linear optimal control. The proposed active approach has similar performance to the classical linear optimal control. However, the classical linear optimal control tends to increase slightly the absolute accelerations compared to the proposed active approach in the attempt to regulate displacements, while the proposed active approach

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