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1 EVALUATING THE EFFECTIVENESS OF STRUCTURAL CONTROL WITHIN THE CONTEXT OF PERFORMANCE-BASED ENGINEERING Luciana R. Barroso 1, Scott E. Breneman 1, and H. Allison Smith ABSTRACT A preliminary study is presented which addresses the role of active control technology in the improvement of structural performance during earthquakes. Performance-based evaluation of control methods is an important step in relating the benets of these non-traditional structural solutions to the earthquake engineering community. A nonlinear model of a steel moment frame is developed and integrated with an active control system. The nonlinear model is a strong-column-weak-girder system, with lumped plasticity model described by the smooth-varying Bouc-Wen hysteretic model. The steel moment resisting frame with tendon actuator in the rst story is subjected to four dierent earthquakes and varying control eort. The results from the control simulations are then presented. 1 Research Assistant; Dept. of Civil Engineering, Stanford Univ., Stanford, CA Associate Professor; Dept. of Civil Engineering, Stanford Univ., Stanford, CA

2 Introduction In recent years, research in the development of control systems has made signicant progress in the reduction of the overall response of civil structures subjected to seismic excitations. However, much of this research has utilized highly simplied linear models of structural systems. To address the broader role of control technology in improving the overall performance of structures, the control analyses presented here consider more sophisticated structural models and include information about the nonlinear response of individual members. For structural control to gain viability in the earthquake engineering community, understanding the role of controllers within the context of performance-based engineering is of primary importance. In general, control studies in civil engineering can be divided into two categories: those which address serviceability issues and those whose main concern is safety. When serviceability is the main concern, control is used to reduce structural acceleration in order to increase occupant comfort during relatively mild wind or seismic excitations. However, for those controllers developed for stronger excitations, where occupant safety is the main concern, the goal is to improve structural response by reducing peak interstory drift or by increasing energy dissipation. The majority of these studies have dealt mainly with linear systems and analyses. Improvement of structural performance under moderate to severe excitations requires a reduction of damage under dynamic loading, and damage is an inherently nonlinear process. Peak responses alone do not describe the possible damage incurred by the structure as cumulative damage results from several incursions into the inelastic range. As such, the reduction of peak interstory drifts alone is not sucient unless we also have information about the capacity of the structure. One cannot assume a structure will remain linear even under moderate seismic loads. The structural engineering community has been making great strides in recent years to develop performance-based earthquake engineering methodologies for both new and existing construction. Both SEAOC's Vision project (SEAOC 1995) and BSSC's NEHRP Guidelines for Seismic Rehabilitation of Buildings (BSSC 1997) present the rst guidelines for multi-level performance objectives. One of the intents of these provisions is to provide methods for designing and evaluating structures such that they are capable of providing predictable performance during an earthquake. The objective of the research presented here is to evaluate the role of structural control technology in enhancing the overall structural performance under seismic excitations. This study focuses on steel moment resisting frames with active tendon bracing in the rst story. The method of evaluating performance in this study is presented with a discussion on its extension to control analyses. A nonlinear dynamic analysis is utilized to determine the response of a structure to four dierent ground motions. This approach can be extended to several dierent structural control solutions, such as passive and hybrid systems.

3 Performance Evaluation Structural performance is a measure of the damage in a structure. Improvement of structural response, requires a reduction of damage under dynamic loading. In general, this evaluation considers both structural and nonstructural components as well as the contents of the structure. No single parameter will provide all the information required at all performance objectives. Good discussions of performance based design and evaluation can be found in Bertero (199), Cornell (199), and Krawinkler (199). Performance evaluation consists of a structural analysis with computed demands on structural elements compared against specic acceptance criteria provided for each of the various performance levels. One of the rst requirements of performance evaluation is the selection of one or more performance objectives. Since the evaluation relies on analysis rather than experimentation, the criteria should be stated in terms of a response that can be calculated. Both the NEHRP and Vision projects have identied similar performance level denitions with slightly dierent naming conventions. These performance levels and corresponding indicative drift limits are listed in Table 1. Both the peak and permanent interstory drifts are utilized in dening performance levels as an indicator of damage. Peak transient drift serves as an indicator of damage to low strength rigid elements, such as building cladding and partition walls. The permanent drift provides a rough indicator of cumulative damage to structural members. Table 1. General Structural Performance Level Denitions and Indicative Drifts for Steel Moment Frames (BSSC 1997). Level Description Drift a Limit (%) Transient Permanent Collapse Little residual stiness and strength, but load Prevention bearing columns and walls function. Large permanent drifts. Building is near collapse. Life Safety Some residual strength and stiness. Some 1 permanent drift. Building may be beyond economical repair. Immediate Occupancy No permanent drift. Structure substantially retains original strength and stiness. 1 negligible a Indicative of a typical structure when responding within performance level. Depending on the intensity of the ground motion, a dierent performance objective will be desired. According to the expected intensity, the designer must analyze whether achieving the objective will be economically feasible. For frequent events, the designer will probably desire that the structure remain fully operational. For rarer events, ensuring prevention against collapse may be the only realistic goal. Structural control provides an extra mechanism to improve structural performance. For 3

4 maximum eectiveness, minimal control eort would be required to achieve the desired performance goals. In active control, this control eort is measured in terms of the energy and forces required of the actuation system. In this study, the peak absolute force is the representative measure of control eort. Attempting to apply very large control forces may not be physically realizable. Under severe seismic loads, meeting the desired performance objective may be unattainable with a given controller design and architecture. In such cases, a dierent control solution should be investigated. Structural System Problem Denition The response of a three-story, steel moment resisting structure is analyzed to illustrate the preceding concepts. The design of the structure was developed for case studies in the SAC Phase Steel Project under 199 Uniform Building Code requirements. The site for the structure is located in Los Angeles and has been designated as having rm soil conditions. A summary of the structural properties can be found in Table. For detailed information on the structural system, the reader is referred to the paper presented by Krawinkler and Gupta (1998) in this conference. Table. Structure Denition of the North-South Moment Resisting Frame. Story Mass (kg) Girders Columns Section Yield (kpa) Exterior Interior Yield (kpa) Stress Stress 1 78,8 W 33 x 118 8, W 1 x 57 W 1 x 311 3,5 78,8 W 3 x 11 8, W 1 x 57 W 1 x 311 3, ,8 W x 8 8, W 1 x 57 W 1 x 311 3,5 Structure Model The type of structure studied is a steel moment-resisting frame designed according to 199 Uniform Building Code requirements utilizing the strong-column-weak-girder (SCWG) design philosophy. According to SCWG, inelastic behavior at a connection should rst occur in the girders rather than the columns. The analytical model developed localizes the inelastic behavior at the base of the structure and the beam elements. All the story columns are presumed to remain elastic. This model was chosen as it models the basic nonlinear structural behavior with minimal complexity. A nite element model of the structure was developed where an assembly of interconnected elements describes the hysteretic behavior of structural members. The inelastic behavior of the members is taken to be concentrated at the end of girders and beams. Thus each structural member is constructed using a lumped plasticity model with nonlinear rota-

5 y 1 y x 1 θ 1 θ 3 θ θ x Figure 1. Lumped Plasticity Model for Beam Element. tional springs at each end joined by a linear beam-column element, as seen in Figure 1. The hinges are zero length elements with very high initial stiness relative to the beam elements. The inelastic behavior is aected only by the moment and rotation of the spring. Hysteretic Behavior Various hysteretic models for the restoring force of an inelastic structure have been developed in recent years. The model chosen for the nonlinear rotational spring described in the previous section is the Bouc-Wen (Wen 197) smooth-varying hysteretic model. This model includes a number of parameters, allowing a mathematically tractable state-space representation capable of expressing several hysteretic properties. The restoring force, (f R ) i, for a single nonlinear element i may be decomposed into two parts, (f E ) i and (f H ) i, representing the elastic and hysteretic components respectively. The restoring force can then be written with the subscript i implied as: f R = f E + f H = k e r(t) + k h z(t) = k e (x a? x b ) + k h z(t) (1) where k e is the portion of the initial stiness corresponding to constant elastic stiness, and k h is the portion associated with the nonlinear behavior. The variable r(t) is the relative deformation, x a and x b are the absolute displacements at nodes a and b respectively, and z(t) is the corresponding variable introduced to describe the hysteretic component. The force-deformation curve is described by: _z = _r dz dr = _r[a? ( + sgn( _rz))jzjn ] () where A,,, and n are shaping parameters (Wen 197) and the term in square brackets, dz=dr, describes the hysteretic curve. Since we require dz=dr to be unity at small values of z, then A = 1. The yield displacement Y = ( + )?1=n ; taking and as equal, Equation can now be written as: h ni _z = _r 1? :5(1 + sgn( _rz)) (3) z Y Now consider a structure idealized by an n-degree-of-freedom system under a one-dimensional earthquake ground motion. The equation of motion for the system can be expressed as: M x(t) + C _x(t) + K E x(t) + K H z(t) =?M flg u g (t) = F g (t) () 5

6 in which x(t) is a vector containing the displacement of each degree of freedom relative to the ground, and z(t) is a vector containing the corresponding hysteretic information for each element. M is the mass matrix and C is the viscous damping matrix. The ground motion, F g, is found by mapping the horizontal ground acceleration, u g, to the horizontal degrees of freedom through the vector flg and multiplying by?m. As in the single element case, the elastic and hysteretic components of the structural restoring force can be separated such that: F R = F E + F H = K E x(t) + K H z(t) (5) so that the restoring force is a function of both x(t) and z(t). The equation of motion for the system can be written in a nonlinear state-space format, as shown in Equation : < _x(t) = I < x(t) = x(t) : ; =?M?1 K E?M?1 C?M?1 K H 5 _x(t) : ; + M?1 5 F g (t) () _z(t) [dz=dx] z(t) where [dz=dx] is a non-square matrix function found by: [dz=dx] = [dz=dr] B rx (7) and [dz=dr] is a diagonal matrix function of _x(t) and z(t) with entries dz i =dr i found in Equation 3, and B rx is a non-square matrix mapping the displacements x(t) to the relative deformations r(t). The system can now be solved using any of a number of numerical algorithms. Model Verication The nite element model described in previous sections was implemented in MATLAB. The structure was modeled as a -D frame using centerline dimensions and nominal yield stress values. The proportional damping was set to approximately % of critical in the rst two modes. The rotational elements were assigned zero strain hardening. This analytical model was veried by comparing the results from this model with those of the nite element program DRAIN-DX (Allahabadi 1987) for several ground motions. DRAIN-DX is only capable of modeling the restoring force using piece-wise linear curves, which in this case was a bilinear curve. In contrast, the smooth curve from the Bouc-Wen model can provide a more realistic representation of structural behavior. The parameter n determines how closely the model reects an elasto-plastic behavior (see Figure (b) for illustration). Figure (a) shows a comparison of the interstory drift time history for the rst story under the excitation of the Whittier, 1987, ground motion described in Table 3. As expected, the smooth hysteretic model implemented resulted in larger permanent deformations than those from the bilinear hysteretic model utilized by DRAIN-DX, as the smooth model will start yielding earlier. Integration with Control The controller architecture applied to the structure is an active tendon brace located in the center bay of the MRF between the ground and second oor. For this study, the sensors

7 1 First Story Drift Response Whittier, 1987 (la59) x Comparison of Hysteretic Models 1 8 Bouc Wen DRAIN DX 3 Displacement (cm) Moment (N m) 1 1 Bouc Wen, n = 5 DRAIN DX Time (sec) Rotation (radians) (a) First Story Drift Figure. Comparison of Analytical Models. (b) Hysteretic Models readings available to controllers are from accelerometers measuring the horizontal acceleration of the oors of the structure. The actuator inuence is idealized by the application of a lateral force onto the beam-column connections of the second oor. The controller algorithms integrated with the above analytical model are linear dynamic output feedback controllers of the form: _x c (t) = A c x c (t) + B c y(t) (8) u(t) = C c x c (t) + D c y(t) (9) where x c (t) is the state vector of the dynamic controller, y(t) is the sensor reading vector, u(t) is the controller command signal, and the constant matrices A c, B c, C c and D c are the linear state description of the regulator. The controller command signal u(t) is mapped to forces applied to the structure by the equation: F u (t) = B u u(t) (1) which is an additional force input to the state equations with the same mapping as F g in Eqn.. The sensor measurements are described by mapping the states to the absolute accelerations at the sensor locations y(t) = C y (?M?1 C _x(t) +?M?1 K E x(t) +?M?1 K H z(t) + F u (t)) (11) where C y selects which accelerations are available. Equations 8-11 are combined with the uncontrolled structure state space equation in Equation to form the controlled system with the augmented state vector x s (t) = fx _x z x c g T with the additional states of the controller 7

8 added to the state vector. The controller loop from the sensors to the actuator forces is algebraically closed to form a state space equation: _x s (t) = A(x s (t)) x s (t) + B w F g (1) which is used to perform the nonlinear time history analysis. The controller architecture presented is the linear dynamic output feedback canonical form with no actuator saturation implemented. For this initial investigation into performance evaluation of controllers, other nonlinear controller algorithms, including nonlinear controllers, are not considered. The application of linear controllers to active control of structures under earthquake loading has received signicant study. One of the aims of this work is to evaluate the eectiveness of this class of controllers while considering the nonlinear behavior of structures under moderate earthquake loading. The limitation of not including actuator saturation will be addressed by creating controllers with reasonable peak control forces. Controller Design Numerical Simulations The controller algorithms, dened by A c, B c, C c, and D c, are created using a H1 approach with the nominal linear structural model as the basis of the design model (Doyle et al. 1989). The ground motion acceleration is weighted using a Kanai-Tajimi lter with the dominant frequency content centered at a period of.31 seconds, a shaping factor of.5 and a RMS gain from unit white noise input of.1 g. In addition, a noise disturbance is added to the sensor acceleration measurements of the design model with a RMS magnitude of the sensor noise from a unit white noise input of.1 g. The controllers are designed to regulate the absolute horizontal accelerations of the three oors. Regulating structural accelerations has been shown to be eective in reducing accelerations and interstory drifts of linear structures in earlier studies by Dyke et al. (199) and Breneman and Smith (1998). The full regulated output for the controller design is of the form: z(t) = [ d 1 (t) + d (t) + d 3 (t) + ] u(t) (13) where d i is the total horizontal acceleration of the i th oor and is a weighting factor on the cost of control eort. The value of used for controller design ranges from.3 to.9, which corresponds to small to large control eort. Using the full order dynamic model of the structure and the control eort design weightings results in a 1th order dynamic model. The H1 dynamic output feedback controllers are designed using a 1 th order model created by the reduction of the full-order state model by balanced-reduction truncation methods. Design models smaller than 1th order signicantly vary from the full order model in the frequencies below 1 Hz. The numerical results presented below are for 1th order dynamic output feedback controllers designed by varying the control eort weight. 8

9 Loading The ground motions selected for the analyses were taken from the sets developed for the SAC Phase Steel Project. These sets contain acceleration records of two horizontal components from 1 dierent earthquakes. The acceleration histories have been scaled so as to conform roughly to the 1997 NEHRP design spectrum for soil type Sd (rm) for a specied return period. Each original ground motion was initially resolved into a fault-normal and a fault-parallel orientation. After manipulation, the components were then rotated 5 degrees from the fault-normal and fault-parallel orientations (Somerville 1997). Four records, two components of two dierent earthquakes, were utilized in this study. Table 3 provides a description of the four acceleration records used in this study, including the linear spectral acceleration value, S a, at the rst mode of the structure. These particular four records were chosen as they produced similar drift demands in at least one component direction. Table 3. Ground Motion Parameters. Name a Record Duration PGA Mean Return S a at First (sec) (cm/sec ) Period (years) Mode (cm/sec ) la15 Northridge, la1 Northridge, la59 Whittier, la Whittier, a Naming convention utilized by the SAC Phase II Project Results Controllers were designed for several dierent values of actuator eort by varying the parameter, and the closed loop state space system was solved using the function ode15s from the MATLAB ode suite. This algorithm is a variable-order method for sti dierential equations utilizing a variable time step of integration. Plots showing the results of these analyses are given in Figures 3-. The uncontrolled response has peak drifts of around % of the story height and some level of residual drift under all earthquakes. Under the Whittier la59 record, the application of control reduces all of the response parameters being monitored. With a control force of less than 1% of the seismic weight (17 kn), the residual drift is reduced to less than 1% in all three stories. A similar behavior is observed for the la record for the peak drifts and residual rst story drift. However, performance improvement quickly reaches a plateau where very little is gained by applying more control force. In the Northridge earthquake simulations, even very high amounts of actuator eort cannot keep the structure from going nonlinear. In both the la15 and la1 simulations, the controller sacrices the rst oor drift for benets in drift reduction of the nd and 3 rd stories for high control force application. This phenomenon is illustrated in Figures 7 and 8 for the la1 record. The only substantial dierence in the response due to the la15 record from the 9

10 Performance Quantities for Nominal H Controller Performance Quantities for Nominal H Controller 7 Whittier, 1987 (la59) 7 Whittier, 1987 (la) Maximum First Story Drift Maximum First Story Drift Maximum Second Story Drift Maximum Third Story Drift Maximum Second Story Drift Maximum Third Story Drift Residual First Story Drift Residual First Story Drift 5 Residual Second Story Drift Residual Third Story Drift 5 Residual Second Story Drift Residual Third Story Drift Percent Story Drift 3 Percent Story Drift Peak Control Force (kn) Peak Control Force (kn) Figure 3. Response Comparison for la59. Figure. Response Comparison for la. 7 Performance Quantities for Nominal H Controller Northridge, 199 (la15) 7 Comparison of Response Quantities for Nominal H Controller Northridge, 199 (la1) 5 Maximum First Story Drift Maximum Second Story Drift Maximum Third Story Drift Residual First Story Drift Residual Second Story Drift Residual Third Story Drift 5 Maximum First Story Drift Maximum Second Story Drift Maximum Third Story Drift Residual First Story Drift Residual Second Story Drift Residual Third Story Drift Percent Story Drift 3 Percent Story Drift Peak Control Force (kn) Peak Control Force (kn) Figure 5. Response Comparison for la15. Figure. Response Comparison for la1. la1 is that the la15 response is characterized by only one inelastic pulse. Also, for the la1 response, the residual drift appears to suer at high control eorts even though the peak drifts are being reduced. Unlike previous records, this drift response is characterized by two large pulses, occurring in opposite directions, which drive the structure into the plastic range. Though the controller does not reduce the rst pulse, it is eective at reducing the second pulse. As the structure is by then oscillating around a new equilibrium point, the reduction of the response at that point results in a larger residual drift. This behavior illustrates one of the shortcomings of using residual drift as a measure of cumulative damage. In this case, the application of control results in less excursions into the plastic range, indicating less damage; however, looking only at the residual drift values would lead to the opposite conclusion. The same conditions explain why the residual drifts become slightly worse in the second and third stories of the la record (see Figure ). 1

11 1 Comparison of First Story Drift LA1 Comparison of Third Story Drift LA1 1 Maximum Control Force = kn Uncontrolled Drift (cm) Drift (cm) 5 Maximum Control Force = kn Uncontrolled Time (sec) Time (sec) Figure 7. First Story Drift for la1. Figure 8. Third Story Drift for la1. Conclusions and Future Work This study presents a preliminary look at the evaluation of active control using a performancebased approach. Performance-based evaluation of control methods is an important step in relating the benets of these non-traditional structural solutions to the earthquake engineering community. The approach taken here shows the eects of one control algorithm on the performance metrics of peak and residual drifts. However, careful inspection of the structural behavior also demonstrates the need for more advanced measures of damage, as peak and residual drifts alone can be misleading. This framework for evaluation of structural control is being expanded to more sophisticated structural models and performance indices. In the future, the expanded framework will be applied to the evaluation of other control algorithm and architectures, including passive and hybrid systems. References Allahabadi, Rakesh. DRAIN-DX, Seismic Response and Damage Assessment of D Structures, Ph.D. Thesis, University of California{Berkeley, Berkeley, CA, Bertero V. V. \The Need for Multi-Level Seismic Design Criteria."Proc. Eleventh World Conference on Earthquake Engineering (June 199) Paper no. 1. Breneman, S. E. and H. A. Smith. \Design of H1 Output Feedback Controllers for the AMD Benchmark Problem."Earthquake Engineering and Structural Dynamics - Structural Control Benchmark Problems. Accepted for publication (1998). BSSC. NEHRP Guidelines for the Seismic Regulation of New Buildings and Other Structures. Second Ballot Version. April, Cornell, C.A. \Calculating Building Seismic Performance Reliability; A Basis for Multi- 11

12 level Design Norms."Proc. Eleventh World Conference on Earthquake Engineering (June 199) Paper no. 1. Doyle, J. C., K. Glover, P. P. Khargonekar, and B. A. Francis. \State-Space Solutions to Standard H and H1 Control Problems."IEEE Transactions on Automatic Control 3, no. 8 (1989): Dyke, S., B. F. Spencer, P. Quast, M. K. Sain, D. C. Kaspari, and T. T. Soong. \Acceleration Feedback Control of MDOF Structures."ASCE Journal of Engineering Mechanics 1, no. 9 (199): Krawinkler, H. \ A Few Basic Concepts for Performance Based Design." Proc. Eleventh World Conference on Earthquake Engineering (June 199) Paper no Krawinkler, H. and A. Gupta \ Story Drift Demands for Steel Moment Frame Structures in Dierent Seismic Regions." th U.S. National Conference on Earthquake Engineering, (1998). SEAOC.Vision - A Framework for Performance Based Design, Volumes I, II, III. Structural Engineers Association of California, Vision Committee, Sacramento, California, Somerville, P. Draft Report. Internal SAC Communication (1997). Wen, Y. \Method for Random Vibration of Hysteretic Systems." Journal of the Engineering Mechanics Division 1, no. EM (April 197):