Practical Design of MDOF Structures With ...

8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability

PMC2000-005

PRACTICAL D ESIGN OF MDOF STRUCTURES WITH SUPPLEMENTAL VISCOUS D AMPERS U SING MECHANICAL LEVERS Y. Ribakov and J. Gluck, M. ASCE Technion - Israel Institute of Technology, Haifa, 32000, Israel [email protected], [email protected] N. Gluck Negev Academic College of Engineering, Beer Sheva, Israel [email protected] Abstract A method for the design of a passive control system for multistory structures is presented. Viscous dampers that are installed at each story of the building are used to improve the response of the structure during earthquakes. Optimal control theory is used to obtain the properties of the devices. The optimization leads to different levels of damping at each story, which is inconvenient and can be expensive. A method that enables the use of devices that are available off-the-shelf is proposed. Numerical analysis of a seven story shear framed structure is represented as an example. It is shown that by using the proposed method the response of structures with devices that are available off-theshelf is close to that of structures with viscous dampers, selected according to the optimal control theory. Significant improvement was obtained in the behavior of the controlled structure compared to the uncontrolled one.

Introduction Viscous dampers are known as effective devices improving structural response to earthquakes. The resulting damping force developed by the viscous damper depends on the physical properties of its fluid, on the pattern of flow in the device and on its size (Constantinou et al., 1992). Gluck et al. (1996) demonstrated that structures dominated by a single mode of vibration, provided with optimally designed supplemental viscous dampers, have a response close to that of an actively controlled structure. An Optimal Control Theory (OCT) using a linear quadratic regulator is adapted to design linear passive viscous devices according to their deformation and velocity. The main difficulty is that the optimal solution requires different levels of damping at each story, which is inconvenient and can be expansive. This paper proposes a method of damper installation by which damping devices that are available off-the-shelf are used at all stories of the structure. According to this method, the damper is connected to a lever arm which amplifies the piston’s displacement and velocity and controls the damping level at each story. Optimal Design and equivalent lever arm approach The response of a structure provided with supplemental dissipating devices is described by the following dynamic equation of equilibrium (Soong, 1990):

Mu˙˙(t ) + Cu˙(t ) + Ku(t ) = Lfe (t ) + Dfc (t ) Ribakov, Gluck J. and Gluck N.

(1)

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where M, C, and K are the mass, damping, and stiffness matrices, respectively; u(t), u˙(t ), and u˙˙(t ) are the displacement, velocity and acceleration vectors, respectively; fc is the vector of forces in the supplemental devices, fe is the external excitation vector, and D and L are the control and excitation forces-location matrices, respectively. The system of second order differential equations (1) may be simplified by a transformation into a state space form:

z˙(t ) = Az(t ) + Bfc (t ) + Hfe (t )

(2)

where z(t ) = [u(t ), u˙(t )] is the vector of the displacements and velocities at each degree of freedom of the structure; A is the system’s matrix, B and D are the location matrices specifying, respectively, the locations of controllers and external excitations. T

 0 A2 n × 2 n =  −1 − M K

 −1  , − M C 1

 0  H2 n × r =  −1   M L

 0  B2 n × m =  −1  ,  M D

(3)

For the case of linear control forces they can be written as follows:

[

]

fc(t) = Gz(t) = G u , G u˙ z(t) = Gu u (t ) +

Gu˙ u˙ (t )

(4)

where G is the gain matrix. The equation of motion reduces to

z˙(t ) = Ac z(t ) + Hfe (t )

(5)

where the matrix of the controlled system, Ac = A + BG. The matrix, G, can be determined from the minimization of the performance index, defined as: tf

J=

∫ [z

T

]

(t )Qz(t ) + fc T (t ) Rfc (t ) dt

(6)

0

where t f is the total time of the considered dynamic event, Q is a 2n × 2n positive semidefinite matrix, and R is an m × m positive definite matrix. Q and R are weighting matrices, representing the relative importance of the state variables and of the control forces in the minimization procedure (Soong, 1990), and

G = −0.5 R −1 BT P

(7)

where P is the solution of the Ricatti algebraic equation AT P + PA − 0.5 PBR−1 BT P + 2Q = 0

(8)

The stiffness and damping are assumed independent, and the damping coefficients of the supplemental devices in a structure with one dominant mode are obtained as follows (Gluck et al. 1996):

  ∆ci =  ∑ gij , d˙ Φ jm  / Φ im  j 

Ribakov, Gluck J. and Gluck N.

(9)

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where Φ im is the element of the eigenvector corresponding to mode m and degree of freedom i and gij , d˙ are gain coefficients in the terms of drift velocities. The damping coefficients are independent on the earthquake history, but only on the characteristics of the structure. The supplemental viscous dampers control the drift by means of a lever fixed by a fulcrum, which is rigidly attached to the upper floor of the story (Fig. 1c). The velocity increment in the supplemental device increases its force output. This setup with levers enables the use of a single device or of a limited variety of supplemental devices in the structure, and maintains the response of the system, under the same dynamic excitations, equivalent to that of the structure with devices selected according to the OCT procedure.

a2

a2

b2 b2

a1 a1 b1 b1

(a)

(b)

damper lever arm (calculated for each story)

chevron fulcrum chevron

rigid lever

(c) Figure 1. A typical Viscous damped structure: (a) controlled by dampers selected according to OCT, (b) controlled by off-the-shelf devices using lever arms, (c) details of damper attachment for the case when lever arms are used.

The force-displacement behavior of the viscous devices has an elliptic shape. The energy dissipated in the device is equal to the area of the ellipse. The peak force (Fdi ) in the linear viscous device is:

a = Fdi = Ci u˙i

(10)

where Ci is the damping coefficient of the device located at the i th story that is set by & optimal control theory, and ui is the peak velocity of the device placed at the i th story. The

Ribakov, Gluck J. and Gluck N.

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lever, presented above, enables the modification of the velocity and of the displacement transferred to the viscous device such that the energy dissipated in it is equal to that of a different, OCT designed device which would have been connected directly to the upper floor of the structure (Fig. 1a). The ratio between the damping coefficient of an OCT designed device, Ci , and that of the device connected to lever arm at the i th story, Ci , is:

RAi = Ci / Ci

(11)

The force developed by the device can be further written as:

Ci Fvd = Ci u˙ = u˙ − i RA i − i

(12)

Requiring that the energies dissipated by the devices that are selected according to OCT and by the devices that are connected to lever arms, are equal, yields:

(

)(

)

Ei = π (Ci / RA i ) u˙ LAi u LAi = π Ci u˙ u −i −i −i −i

(13)

LAi = ( ai + bi ) / bi

(14)

where

is the lever arm ratio required to maintain the equivalency of the energies in the two cases and ai , bi are shown in Fig. 1(b). The solution of Eq. (13) leads to LAi = RAi

(15)

which is the relationship between the lever arm ratio, LAi, and the ratio of the damping coefficients, RA i . Numerical example To investigate the effectiveness of the proposed design technique simulations of a sevenstory building were carried out. A shear framed structure, with stiff beams was chosen. The responses to four different seismic excitations were computed: The following four seismic excitations were used as input in the analysis: El-Centro S00E, 1940, Taft N21E, 1952, Loma-Prieta N90E, 1989, and Eilat EL1226NS, 1995. The structure was characterized by the following matrices: M = 87500 × I 7 × 7 [kg - mass], 0   29.28 −14.64    − 14.64 31.59 − 16.95   −16.95 30.96 − 14.01   7 K= − 14.01 28.02 −14.01  ×10 [N / m]   − 14.01 25.13 − 11.12   − 11.12 22.24 −11.12    0 − 11.12 11.12  

Ribakov, Gluck J. and Gluck N.

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where M is the mass matrix of the structure, I7×7 is a unit diagonal matrix, and K is the structural stiffness matrix. An initial 1% damping ratio was assumed for the first vibration mode of the uncontrolled structure. The optimization was carried out using the OCT implemented in a MATLAB routine, which yields the following optimal values of the viscous coefficients: N × sec ∆Ck = [1.7219 1.7219 1.9936 1.6478 1.6478 1.3079 1.3079] × 10 6 m The matrix represents the optimal solution of a structure provided by supplemental viscous dampers. The solution requires different damping levels at each story, which may be significantly simplified by application of the lever arm technique, with C6 = C7 = 1.07×10 6 N × s / m, and Ci =1.6×10 6 N × s/m, at floors 1-5. These dampers are connected to lever arms, and according to the proposed procedure the lever arm ratio vector T is: LAi = [1.076 1.076 1.114 1.013 1.013 1.106 1.106] . All simulations were performed the ETABS 6.2 package (Habibullah 1994). Peak displacements and base shear forces for the uncontrolled, optimally designed structure and for the structure with devices connected to lever arms, respectively, are presented in Tables 1, 2 and 3. Tables 2 and 3 show that the solution with the levers yields very similar results to those of the optimal solution. Using the proposed technique the peak displacements and base shear forces in the controlled structure were reduced up to 50% (see Tables 1, 2, 3).

Displacements, cm

Story 7 6 5 4 3 2 1

Base shear, kN

El - Centro 4.33 4.02 3.51 2.96 2.29 1.65 0.84 1220.7

Taft 5.18 4.80 4.17 3.48 2.66 1.90 0.96 1383.0

Loma - Prieta 3.48 3.14 2.57 1.96 1.60 1.43 0.91 1334.8

Eilat 4.18 3.83 3.27 2.71 1.06 1.48 0.75 1139.6

Table 1. Peak response of the uncontrolled structure.

Displacements, cm

Story 7 6 5 4 3 2 1

Base shear, kN

El - Centro 2.76 2.61 2.36 2.00 1.54 1.17 0.60 900.9

Taft 2.39 2.23 1.95 1.63 1.25 0.89 0.45 663.7

Loma - Prieta 2.03 1.84 1.54 1.23 0.89 0.65 0.38 624.9

Eilat 2.68 2.50 2.19 1.84 1.43 1.04 0.54 810.1

Table 2. Peak response of the structure controlled by dampers selected according to OCT.

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Displacements, cm

Story 7 6 5 4 3 2 1

Base shear, kN

El - Centro 2.85 2.68 2.38 2.03 1.59 1.15 0.59 853.8

Taft 2.62 2.45 2.17 1.79 1.33 0.99 0.50 802.9

Loma - Prieta 2.14 1.97 1.69 1.33 0.97 0.70 0.34 609.9

Eilat 2.67 2.50 2.22 1.84 1.37 1.01 0.50 739.3

Table 3. Peak response of the structure controlled by off-the-shelf dampers connected to lever arms.

Conclusions A procedure was developed for optimal design of passive controlled viscous damped structures with devices connected to lever arms. The viscous properties of the devices were selected using the OCT design, and the lever arm approach was then used to find the lever arm ratios for application of devices that are available off-the-shelf. A numerical simulation of a seven-story structure showed that its performance with viscous devices connected to lever arms is close to that of an optimally designed one. Reductions of up to 50% of the peak displacements and of the base shear forces were obtained. These results show a promising method of using off-the-shelf available viscous dampers by connecting them to lever arms. Thus, the damping level at each floor is controlled and provides an improved response of the structure during an earthquake. References Constantinou, M. C., and Symans, M. D. (1992), “Experimental and analytical investigation of seismic response of structures with supplemental fluid viscous dampers,” Tech. Rep. NCEER-920027, National Center of Earthquake Engrg. Res. State Univ. of New York (SUNY) at Buffalo N.Y. Gluck, N., Reinhorn, A. M., Gluck, J., and Levy, R. (1996). “Design of supplemental dampers for control of structures,“ ASCE Journal of Structural Engineering, 122(12), 1394-1399. Soong, T. T. (1990), “Active structural control: theory and practice,” John Wiley & Sons, Inc., New York, N.Y. Habibullah, A. (1994), “ETABS Three dimensional analysis of building systems, user’s manual, version 6.0,” Computers and Structures Inc., Berkeley, California.

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