benchmark problem for control of base isolated ...

BENCHMARK PROBLEM FOR CONTROL OF BASE ISOLATED BUILDINGS Sriram Narasimhan 1 , Satish Nagarajaiah 1 , Member, ASCE, Henri Gavin2 , Member, ASCE, and Erik A. Johnson 3 , Assoc. Member, ASCE ABSTRACT This paper presents the benchmark problem definition for seismically excited base-isolated buildings. The objective of this benchmark study is to provide a well defined base isolated building with a broad set of carefully chosen parameter sets, performance measures and guidelines to the participants, so that they can evaluate their control algorithms. The control algorithms may be passive, active or semi-active. The benchmark structure considered is an eight story base isolated building similar to existing buildings in Los Angeles, California. The base isolation system includes both linear and non-linear bearings and devices. The superstructure is considered to be a linear elastic system with lateral-torsional behavior. A new nonlinear dynamic analysis program has been developed and made available to facilitate direct comparison of results of different control algorithms.

Keywords: Benchmark, Nonlinear, Base Isolated Building, Control Algorithms. INTRODUCTION Base isolation systems, such as sliding and elastomeric bearing systems, reduce the superstructure response, but with increased base displacements. In the case of near-fault ground motions, which are characterized by strong impulsive motion, excessive displacements can occur resulting in the failure of the isolation bearings. Current practice is to provide non-linear passive dampers to limit the bearing displacements, however, this increases the forces in the superstructure and also at the isolation level. In order to reduce these excessive motions of the base, active or semi-active devices are adopted. Some of the commonly used isolation systems are elastomeric isolation systems and sliding isolation systems. In buildings, sliding isolation systems support and decouple the superstructure from ground. Sliding bearings consist of Teflon - stainless steel, flat or spherical, interface. Sliding bearings dissipate energy due to frictional behavior. Elastomeric isolation systems consist of laminated rubber bearings reinforced with steel plates. Energy dissipation capacity is provided by a lead plug within the rubber unit, as in lead-rubber bearings, or by inherent damping capacity of the rubber. 1

Graduate Student and Associate Professor, Dept. of Civ. and Env. Engrg., Rice Univ., 6100 S. Main Street, Houston, TX 77005. E-mail: [email protected], [email protected] 2 Assistant Professor, Dept. of Civ. and Env. Engrg., Duke Univ., Durham, NC 27708. E-mail: [email protected] 3 Assistant Professor, Dept. of Civ. and Env. Engrg., Univ. of Southern California, Los Angeles, CA 90089. E-mail: [email protected].

1

TABLE 1. Floor Masses and Eccentricities Floor

1

2

3

4

5

6

7

8

Mass

176

154

141

141

139

105

98

73

-7.8 8.7

-7.8 6.4

-7.9 -6.6

-6.6 -11.3

-6.6 -26.5

1.4 -32.8

11.3 -27.6

9.1 -30.8

(kips−sec2 /f t)

Eccent. (f t)

ex ey

TABLE 2. Modal Periods Mode Period (secs)

N-S 1 2 3 0.76 0.31 0.17

E-W 1 2 3 0.89 0.37 0.22

T 1 2 3 0.67 0.25 0.17

Extensive research has been performed both analytically and experimentally for active, semi-active and hybrid control systems in civil engineering applications. Active control has been implemented in many structures recently. Semi-active control of linear and non-linear structures using novel devices such as Magneto-Rheological (MR) dampers, Electro-Rheological dampers and variable stiffness systems has gained significant attention in the recent years. The effectiveness of structural control strategies and different control algorithms has been demonstrated, by many researchers, experimentally and analytically. In order to provide a common basis for comparing different controllers a series of benchmark problems for fixed base building structures have been developed and studied widely (Ohtori et al. 2001 and Yang et al. 2000). Participants of this benchmark study can propose control strategies for the base isolated building model and shall define, evaluate, and report the results for the proposed strategy. A set of evaluation criteria have also been developed for the sake of comparison of various control strategies. BENCHMARK BASE ISOLATED STRUCTURE The structure is a base-isolated eight-story, steel-braced framed building, 270.4-ft long and 178-ft wide. The floor plan is L-shaped as shown in Fig.1. The benchmark structure considered is an eight story base isolated building similar to existing buildings in Los Angeles, California. A detailed 3D model of the fixed-base superstructure is developed with a rigid floor slab assumption. The building has setbacks after the fifth floor. The superstructure bracing is located at the building perimeter. Metal decking and a grid of steel beams support all concrete floors. The steel superstructure is supported on a reinforced concrete slab, which is integral with concrete beams below, and drop panels below each column location. The isolators are connected between these drop panels and the footings below. The seismic isolation system consists of 61 spherical sliding bearings and 31 linear elastomeric bearings as shown (Fig. 1). The superstructure is modeled as a linear elastic system. The superstructure members, such as beam, column, bracing, and floor slab are modeled in detail (Fig.2). Three master DOF at the center of mass of each floor are used in the condensed model. Hence, only 24 DOF are retained in modeling the superstructure using fixed base properties. The model including the superstructure and isolation system consists of 27 DOF. The mass of each floor and eccentricities between the center of mass and center of stiffness at each floor, are shown in Table 1. The computed periods Tn for the first nine modes in the fixed-base condition are shown in Table 2. The damping ratio is assumed to be 5% in all modes. 2

FIG. 1. Building Plan, Bearing Locations and Elevation (All units in feet: 1 m = 3.28 ft)

THREE DIMENSIONAL NONLINEAR DYNAMIC ANALYSIS Base isolated buildings are designed such that the superstructure remains elastic. Hence, in this study the superstructure is modeled by a condensed linear elastic system. Also, the localized nonlinearities at the isolation level allow condensation of the linear superstructure. In addition, the equations of motion are developed in such a way that the fixed-base properties are used for modeling the linear superstructure. The isolation system is modeled using linear or nonlinear elements representing bearings, dampers or hydraulic devices, described later. The base and the floors are assumed to be infinitely rigid in plane. The superstructure and the base are modeled using 3 master degrees of freedom (DOF) per floor at the center of mass. Each nonlinear isolation bearing or device is modeled explicitly using discrete biaxial Bouc-Wen model, and the forces in the bearings or devices are transformed to the center of mass of the base using a rigid base slab assumption. The equations of motion for the super-structure and the base are as follows: ´ ³ ¨b ¨ + CU ˙ + KU = −MR U ¨g + U (1) MU ´i ³ ´ h ³ ˙ b + Kb Ub + f + f c = 0 ¨g + U ¨ +R U ¨g + U ¨ b + Mb U ¨ b + Cb U RT M U

(2)

where M,C and K = superstructure mass, damping, and stiffness matrices, respectively, in ¨ U, ˙ and U represent the floor acceleration, the fixed-base condition; R = influence matrix; U, ¨ b = vector of base acvelocity and displacement vectors, respectively, relative to the base; U ¨ celeration relative to the ground; Ug = vector of absolute ground acceleration; Mb = diagonal 3

FIG. 2. Detailed Three Dimensional Superstructure Model. mass matrix of the rigid base; Cb = resultant damping matrix of viscous isolation elements; Kb = resultant stiffness matrix of linear elastic isolation elements and f = vector containing the forces mobilized in the isolation bearings and devices, and fc = control forces. Combining Eq.(1) and (2) and using modal transformation U = φU0 (φ = modal matrix of the fixedbase superstructure, normalized with respect to the mass matrix; and U0 = modal displacement vector relative to the base), the following equation is obtained ¨ c +Cc U ˙ c +Kc Uc +Fc = BE U ¨ g = Pc Mc U (3) ½ 0 ¾ · T · ¸ ¸ φ Mφ φT Cφ 0 φT MR U where Uc = ; Mc = ; Cc = ; RT Mφ RT MR + Mb 0 Cb Ub · T ½ · ¸ ¾ ¸ φ Kφ 0 0 φT MR ; Fc = ; BE = − . Kc = f + fc 0 Kb RT MR + Mb in which the following diagonal matrices are obtained because the modal matrix φ is normalized with respect to the mass: φT M φ = I, φT Kφ = ω 2 , and φT Cφ = 2ζω, with ω = diagonal matrix of natural frequencies of the fixed base structure and ζ = diagonal matrix of damping ratios of the fixed base structure. Eq.(3) can also be written in incremental form, at time t + ∆t (∆t = time step), as ˙ c +Kc ∆Uc +∆Fc = ∆Pc ¨ c +Cc ∆U M∆U Using X =

A=

·

½

Uc ˙c U

¾

(4)

, and Eq.(3), the state space equations can be formulated as

´ ³ ˙ (t) = AX (t) +Bc Fc (t) +EU ¨ g (t) = g X, Fc , U ¨g X

0 I −1 K −M −M−1 c c c Cc

¸

, Bc =

·

0 −M−1 c 4

¸

,E =

·

0

−M−1 c BE

(5) ¸

.

The Eq.(5) is solved using unconditionally stable Newmark’s constant-average acceleration method, which can also be derived from trapezoidal rule given by ∆t (gk + gk+1 ) (6) Xk+1 = Xk + 2 ¡ ¢ where gk+1 = g Xk+1 , Fck+1 , ugk+1 ; hence, the method is implicit, needing iteration. In incremental formulation iterations can be avoided. The forces, f, mobilized in the elastomeric isolation bearings or devices can be modeled by a elasticviscoplastic model with strain hardening fx = kp Ux + cv U˙ x + (ke − kp ) U y zx

(7)

fy = kp Uy + cv U˙ y + (ke − kp ) U y zy

(8)

where ke = pre-yield stiffness, kp = post-yield stiffness, cv =viscous damping coefficient of the elastomeric bearing or device, U y is the yield displacement, and zx and zy are dimensionless hysteritic variables (Park et al.1986). ³ ³ ´ ´ ³ ³ ´ ´   ½ ¾ ¾ 2 γ sgn U ˙ ˙ z z z + β z γ sgn U + β z x x x y y y x z˙x U˙ x y ³ ³ ´ ´ ³ ³ ´ ´   =a − U z˙y U˙ y zy2 γ sgn U˙ y zy + β zx zy γ sgn U˙ x zx + β (9) in which parameters a, β and γ are dimensionless quantities, Ux , Uy and U˙ x , U˙ y , represent the displacements and velocities respectively, that occur at the isolation bearing or device. ½

¾

½

U˙ x U˙ y

Eq. (7), (8) and (9) can also be used to model sliding bearings with flat or spherical sliding surface, by means of a small yield displacement U y (because of rigid plastic behavior and large pre-yield stiffness) setting cv = 0 and (ke − kp ) U y = µN fx = kp Ux + µN zx

(10)

fy = kp Uy + µN zy

(11)

where µ is the coefficient of friction and N is the average normal force at the bearing (normal force variation is neglected). In a similar manner other devices such as nonlinear fluid dampers can also be modeled using Eq. (7), (8) and (9). Eq.(9) is solved using the unconditionally stable semi-implicit Runge-Kutta method (Rosenbrook 1964) suitable for the solutions of stiff differential equations. Eq’s. (5), (7), (8) and (9) are solved using the efficient predictor-corrector algorithm and verified using 3D-BASIS program (Nagarajaiah et al. 1991 a, b). MATLAB IMPLEMENTATION The analytical model is implemented using MATLAB and SIMULINK as shown in Fig.(3)(4). The analysis program comprises of a data file, a preliminary file to assemble the data into system matrices for input into the non-linear analysis block 3D-BASIS-SAIC, which is a SIMULINK based S function program as shown in Fig. 3. The full implementation procedure 5

input Structure Data input

Input Data

Data

Form System Matrices

Output Data

Nonlinear Dynamic Analysis Simulink Block MATLAB Program Prelim.m

input Control Data

FIG. 3. Schematics for MATLAB Implementation

is also shown in Fig.4. The inputs to the nonlinear analysis block are the seismic excitation and the control forces provided by the control devices. The non-linear response is calculated using a predictor-corrector algorithm. All the sensor and control devices can be modeled in this program as SIMULINK blocks and the outputs of these models fed into the analysis S-function block. Control Design The proposed nominal isolation system has a fundamental isolation period of TI =2.5 sec, with a frictional damping of µ = 10%. The benchmark participants can vary the isolation period, TI , between 1.0 and 5.0 sec; however, to have a common basis for comparison of different control strategies the participants need to perform one set of simulations for the nominal case TI =2.5 sec in addition to their own control simulations. Participants must strongly justify and defend significant deviations from the nominal isolation system. The benchmark study participants, designers/researchers, are to design an appropriate passive, active, or semi-active control strategy, or a combination of them. It is left to the designers/researchers to define the type, appropriate model and location of the sensor device(s)/ sensor(s) and to develop control algorithms. The S function program 3D-BASIS-SAIC will remain invariant to the various control strategies developed and implemented by the participants. The various control strategies can be compared to one another by having the model and evaluation criteria common to all controllers. The analysis model from Eq. (5) can be written as ´ ³ ¨g ˙ = g X, Fc , U (12) X ³ ´ ¨g Y0 = h X, Fc , U

(13)

where, X is the state vector, Y0 is the vector of outputs, which are to be specified in the input data file. The control algorithm is required to take the discrete form ¡ ¢ Xk+1 = gc Xk , Yk , Fck , uk , Ugk uk = hc (Xk , Yk )

(14) (15)

where, Xk is the discrete state vector at time t = kT , u is the control command. The sensors and other measurement devices can be modeled similarly.

6

yo Responses [t', X]

Matrix Selector

J1toJ8

Selected Outputs

Performance Indices

THREE-D-BASIS-SAIC

Earthquake

Matrix Selector

S-Function

1

Demux

z Unit Delay

signal 1

Selected Output

signal 2

yD Demux

Noise1

Device Response

Matrix Selector Demux

Noise2

Mux

signal 1 signal 2

J9

signal out

Sensors for Structure signal out

Sensors for Devices

Performance Index

Device Outputs

signal

Matrix Selector

signal

in

device output

Discrete Controller

selected responses

f and Device Output

control devices

FIG. 4. SIMULINK Block Diagram for Simulation of Control Algorithms

Earthquake Ground Motions Candidate earthquakes that are to be used for the analysis are Imperial Valley-El Centro, Northridge-Rinaldi, Northridge-Sylmar, Northridge-Newhall, Kobe, and Ji Ji-TCU. The bidirectional ground motion for each record will be provided. The vertical ground motion will not be considered. EVALUATION CRITERIA The following evaluation criteria are defined for the benchmark problem. These criteria should be used as a common evaluation tool to benchmark the performance of the control algorithm proposed. 1. Peak Base Shear (at isolation level)/Peak Base Shear for Uncontrolled Structure, J1 (q) = maxkVo (t,q)k t

. maxkVˆo (t,q)k t

2. Peak Structural Shear (at First Story Level)/Peak Structure Shear for Uncontrolled Struc-

ture, J2 (q) =

maxkV1 (t,q)k t

maxkVˆ1 (t,q)k

.

t

3. Peak Isolator Deformation/Peak Isolator Deformation of Uncontrolled Structure, J3 (q)= maxkdi (t,q)k t,i

. maxkdˆi (t,q)k t,i

4. Peak Inter-story Drift/Peak Inter-story Drift for Uncontrolled Structure, J4 (q) =

maxkdf (t,q)k t,f

. maxkdˆf (t,q)k t,i

5. Peak Absolute Floor Acceleration/Peak Absolute Floor Acceleration for Uncontrolled maxkaf (t,q)k t,f Structure, J5 (q) = max a (t,q) . k f k t,i 6. Peak Device Force/Peak Base Shear, J6 (q) =

° ° °P ° max° Fk (t,q)° ° ° t k

maxkV0 (t,q)k t

7

.

7. RMS isolator deformation/RMS isolator deformation for the uncontrolled structure, J7 (q) =

maxkσd (t,q)k i

maxkσdˆ(t,q)k

.

i

8. RMS acceleration/RMS acceleration for the uncontrolled structure, J8 (q)=

9. Total Energy Absorbed by Devices/Energy Input into the Structure, J9 (q) =

maxkσa (t,q)k f

maxkσaˆ (t,q)k . f   P  TRq   Fk (t,q)dk (t,q)dt k

Tq R

0

dt

0

where, i =isolator number, 1,..., Ni (Ni =8); k=device number, 1, ..., Nd (Nd =8); f =floor number, 1,...,Nf ; q =earthquake number: 1,...,6; t = time, 0 ≤ t ≤ Tq ; h·i = inner product; k·k =vector magnitude incorporating NS and EW components. Further details, data files and Matlab and Simulink files can be found at http://www.ruf.rice.edu/~nagaraja/baseisolationbenchmark.htm. ACKNOWLEDGMENTS Funding for this project provided by the National Science Foundation, NSF-CAREER Grant, with Dr. S. C. Liu and Dr. P. Chang as the Program directors, is gratefully acknowledged. REFERENCES Gavin, H.P., (2001), “Control of Seismically-Excited Vibration using Electrorheological Materials and Lyapunov Methods,” IEEE Trans. on Control Sys. Tech., vol. 9, no. 1, pp. 27–36. MATLAB (1994), The Math Works, Inc., Natick, Massachusetts. Nagarajaiah, S., Reinhorn, A.M., and Constantinou, M.C., (1991a), ”3D-BASIS:Nonlinear dynamic analysis of three dimensional base isolated structures-Part II,”Rep. No. NCEER-910005, Nat. Ctr. for Earthquake Engrg. Res., State University of New York, Buffalo. Nagarajaiah, S., Reinhorn, A.M., and Constantinou, M.C., (1991b), ”Nonlinear dynamic analysis of 3-D-base-isolated structures,”J. of Str. Engrg., ASCE, Vol. 117(7), pp. 2035-2054. Narasimhan, S., and Nagarajaiah, S., (2002), ”3DBASIS-SAIC: Nonlinear Dynamic Analysis of smart base isolated structures, ”Mechanics Research Report No.53, Dept. of Civil and Env. Engrg., Rice University. Ohtori, Y., Christenson, R.E., Spencer, B.F., and Dyke, S.J., (2001), ” Benchmark problems in seismically excited nonlinear buildings, ” http://www.nd.edu/~quake, University of Notre Dame. Park, Y.J., Wen, Y.K., and Ang, A.H.S., (1986), ”Random vibration of hysteritic systems under bi-directional ground motions,” Earthquake Engrg. Struct. Dyn., Vol. 14(4), pp. 543-557. Rosenbrock, H.H., (1964), ”Some general implicit processes for the numerical solution of differential equations.” Computer J., Vol. 18, pp. 50-64. Sahasrabudhe, S., and Nagarajaiah, S.,(2001), ”Sliding Isolated Buildings with Smart Dampers," Proc. Structures Congress, ASCE, Washington, D.C., CDROM. Spencer, B. F., Johnson, E. A., and Ramallo, J. C. (2000), ”Smart Isolation for Seismic Control," JSME Int. J., Series C, 43 (3). Yang, J.N., Agarwal, A., Samali, B., and Wu, J.C., (2000), ”A benchmark problem for response control of wind excited tall buildings.” Proc. 14th Eng. Mech. Cong., ASCE, UT Austin, CDROM, http://www.eng.uci.edu/~anil/benchmark.html and http://www-ce.engr.ccny.cuny.edu/people/faculty/agrawal/benchmark.html. 8

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