System Identification and Health Monitoring Studies on Two Buildings in Los Angeles

System Identification and Health Monitoring Studies on Two Buildings in Los Angeles Derek Skolnik, Graduate Student Researcher, UCLA Eunjong Yu, Assistant Professor, Hanyang University, Korea Ertugrul Taciroglu, Assistant Professor, UCLA John W. Wallace, Professor, UCLA Presented at the 6 th International Workshop of Structural Health Monitoring Stanford University, September 11-13, 2007 Student Paper of Distinction, 2 nd Place ABSTRACT Presented are applications of modal system identification and finite element model updating techniques utilizing data collected during earthquake and forced vibration tests of two buildings in Los Angeles. The Louis Factor building, permanently instrumented with a 72-channel accelerometer network, is continuously recording responses to earthquake and ambient vibrations. The Four Seasons building, damaged in the 1994 Northridge earthquake, served as the pilot project of the nees@ucla equipment site, in which hundreds of sensors were deployed to record responses to forced vibrations from linear and eccentric mass shakers. For each structure, modal properties are identified first; and these data are subsequently used to update their initial finite element models. Novel improvements to a sensitivity-based updating method addresses numerical difficulties associated with ill-conditioning. Predictions of updated models compare well with measured data. INTRODUCTION Collaborative studies on system identification and Finite Element Model (FEM) updating are summarized by presenting their application to the Factor and Four Seasons building. A full account of these efforts may be found in [1], and [2]. The 15-story Factor building Figure 1a), located on UCLA campus, was designed and constructed in the late 1970s. The structural system consists of steel Special Moment Frames (SMF) supported by bell caissons and spread footings. A typical floor plan in Figure 1b displays the orientation of the SMF and typical accelerometer distribution. Following the 1994 Northridge earthquake, the USGS instrumented the structure with 72 uniaxial force-balanced accelerometers; four per floor, including two subterranean levels. The sensor network underwent a series of upgrades beginning in 2003, using funds provided primarily by the NSF Science and Technology Center for Embedded Networked Sensing headquartered at UCLA. All 72 channels were rewired and converted to a 24-bit, continuously recording system, in addition to the nearby installation of various borehole seismometers. The level of instrumentation provided in and around the Factor building makes it one of the most densely instrumented 6 th IWSHM, September 2007 1

buildings in North America. As such, it has recorded terabytes of ambient vibration data as well as numerous small earthquakes. One such event occurred on September 28, 2004, due to ground shaking originating from Parkfield, CA (Mw = 6.0). The epicenter was located approximately 163 miles from the Factor building; and a peak roof acceleration of 0.0025g was recorded. Figure 1. Factor building (a) west face and (b) typical floor layout. Figure 2. Four Seasons Building (a) typical floor plan showing sensor and shaker locations, observed damage due to (b) slab punching and (c) joint shear cracking The Four Seasons Building is a 4-story reinforced concrete office building, constructed in 1977 and located in Sherman Oaks, California. The building suffered damaged from shaking during the 1994 Northridge Earthquake. Figure 2a shows the typical floor plan and sensor layout. A SMF constitutes the building perimeter, whereas interior gravity frames consist of post-tensioned slabs and square columns with drop panels. Bell caissons connected by grade beams form the building s foundation. Prior to forced-vibration testing, visual inspections were performed to document the earthquake damage, which included slab punching failures (Figure 2b), significant diagonal cracks in beam-column joint regions (Figure 2c), column 6 th IWSHM, September 2007 2

flexural cracks, and concrete spalling at beam ends adjacent to the beam-column joints. A series of forced vibration tests were conducted using the nees@ucla equipment (advanced wireless data acquisition systems, linear and eccentric mass shakers). Experimental data used for model updating were obtained from a forced vibration test using a linear mass shaker, which was installed on the roof with an angle of 45 degrees from the N-S axis (Figure 2a), and used to apply broadband excitations (i.e., whitenoise and various sine sweep forces). SYSTEM IDENTIFICATION There is a large variety of system identification algorithms in open literature. For the present study, the Numerical algorithm for Subspace State-Space System Identification (N4SID) was adopted [3]. This non-iterative approach yields reliable state space models for complex multivariable dynamical systems, directly from measured data with modest computational effort. A linear time-invariant dynamical system can be described by the discrete first order differential equation in the state space at the k th time step as x k +1 = Ax k + Bu k + ν k, y k = Cx k + Du k + ω k (1) where x is the state vector; A is the state matrix; B is the input influence coefficient matrix; C is the real output influence matrix; D is the output control influence coefficient matrix; u is the observed input; and y is the output vector. In (1), ν and ω represent measurement and process noise terms, respectively, both of which may be modeled as uncorrelated, zero-mean, stationary, white noise vector sequences. In case of ambient vibrations, the unknown input term u is treated as white noise, and thus, is lumped into ν and ω. The N4SID algorithm first estimates the Kalman state sequence (with linear algebra tools such as orthogonal/oblique projections and SVD) and subsequently yields system matrices (A and C) using a linear least-squares approximation. The complex eigenvalues (λ) and eigenvectors (ψ) of the damped system can be calculated from the system matrix A; and if the damping is assumed to be small and nearly classical, then the familiar structural modal properties can be derived as f i = λ i 2π, ζ i = Re λ i ( ) 2π f i, φ i = Cψ i sign Re( Cψ i ) (2) where Re( ) and sign[ ] denote the real part and the algebraic sign of their arguments; and f i, ζ i, and φ i, respectively, are the natural frequencies, damping ratios and the mode shapes for the i th mode. Implementation of the N4SID algorithm requires a priori selection of the order of the state space model (i.e., the dimension of state vector x), which is a challenging problem. In general, an N degree-of-freedom system will have a statespace model order of 2N. However, because of measurement noise and the nonwhite noise nature of the unknown excitation (in the case of ambient vibration data), a model order that is higher than 2N is needed for extracting as many physically meaningful modal parameters as possible. Because of such high model orders, the N4SID algorithm identifies superfluous modes. In order to distinguish the structural modes from the superfluous ones, stability plots (Figure 3) are 6 th IWSHM, September 2007 3

typically employed, which function on the assumption that as the model order increases, the identified structural modes (and hence associated modal properties), should remain reasonably stable. Stability tolerances are chosen based on the change in frequency (Δf) and damping ratios (Δζ), and on the Modal Assurance Criterion (MAC). The final modal properties are selected as the stable modes of the highest model order and are displayed in TABLE I, and TABLE II, as identified. Figure 3. Stability plot using Factor building data. TABLE I. FACTOR BUILDING MODAL PROPERTIES Mode Identified Initial Model Updated Model No Dir f m (Hz) ζ m (%) f i (Hz) f i / f m MAC f u (Hz) f u / f m MAC 1 EW 0.47 5.10 0.51 1.10 1.00 0.47 1.01 1.00 2 NS 0.51 8.30 0.51 1.01 1.00 0.51 1.02 1.00 3 Tor 0.68 10.80 0.67 0.98 1.00 0.69 1.02 0.99 4 EW 1.49 2.10 1.51 1.01 1.00 1.51 1.01 0.99 5 NS 1.67 1.40 1.45 0.87 0.98 1.67 1.00 0.99 6 Tor 2.36 2.90 1.90 0.81 0.98 2.32 0.98 0.98 7 EW 2.68 2.20 2.53 0.95 0.99 2.58 0.96 0.99 8 NS 2.86 1.30 2.39 0.83 0.95 2.76 0.97 0.99 9 Tor 3.83 2.90 3.19 0.83 0.94 3.74 0.98 0.99 TABLE II. FOUR SEASONS BUILDING MODAL PROPERTIES Mode Identified Initial Model Updated Model No Dir f m (Hz) ζ (%) f i (Hz) f i / f m MAC f u (Hz) f u / f m MAC 1 EW 0.88 5.66 0.89 1.01 0.98 0.89 1.01 1.00 2 NS 0.94 6.94 1.08 1.15 0.99 0.96 1.02 0.99 3 Tor 1.26 6.01 1.29 1.02 1.00 1.26 1.00 1.00 4 EW 2.73 5.61 2.64 0.97 0.90 2.72 1.00 0.99 5 NS 2.94 7.69 2.99 1.02 0.94 2.93 1.00 0.98 6 Tor 3.44 6.14 3.42 0.99 0.93 3.44 1.00 0.99 FINITE ELEMENT MODEL UPDATING Initial FEMs were constructed based on architectural/structural drawings and suitable assumptions. As expected, there exists a considerable discrepancy between identified and analytical modal properties (see, TABLE I and TABLE II). Model updating is performed to reduce these discrepancies and hence, to better predict building responses. Herein, we present an iterative sensitivity-based updating 6 th IWSHM, September 2007 4

method utilizing Frequency Response Functions (FRFs) and/or modal properties. In general, model-updating algorithms begin with the definition of a vector of parameters p, to be updated. Here, parameters are based on physical properties that are not readily modeled such as stiffness contribution of non-structural components (NSCs) or stiffness reduction to do structural damage. Secondly, a residual vector r which measures the error between identified modal data (eigenvalues Ω m and eigenvectors Φ m ) and those analytically obtained, which depend on p, is defined; r T (p) = ( ž m ž (p))./ž m, Φ m Φ(p) (3) The goal of model updating is to determine the set of parameters that minimizes an appropriate norm of this error residual. This objective can be cast as a constrained nonlinear least-squares minimization problem as in min p 2 r(p) L2 such that 0 p i 1. (4) One way to solve (4) is to expand the vector of analytical eigen-properties into a Taylor series and truncate it to include only the linear term [2], i.e., ε(p (a), Δp) = r(p) p=p (a ) p r(p) Δp (5) p=p (a ) where Δp = p p (a) is the parameter perturbation vector at the linearization point a. Equation (5) can be expressed in a compact form as ε = r (a) S (a) Δp where S (a) and r (a) are the sensitivity matrix and residual vector evaluated at p = p (a), respectively. Consequently, the constrained nonlinear minimization problem becomes linear which yields an iterative sequence for p; and if the initial guess is reasonably close to the optimum parameter values, this sequence will be convergent. The sensitivity matrix can be numerically evaluated for a given iteration using the forward-difference method. However, this matrix is often ill-conditioned because groups of distinct parameters may have very similar influences on the error residuals. This is a complication that arises frequently in inverse problems, and occurs, essentially, when the identification of a larger number of parameters is attempted than that warranted by the quantity and the information content of experimental data. The ill-conditioning may lead to a solution that contains large changes in some parameters and small changes in others, which may not be physically sound. In order to address this issue, an additional parameter constraint is introduced, based on correlation coefficients between all the parameter sensitivities, computed as, p i p j R ij where R ij = 1 C ij C ii C jj (6) where C is the covariance matrix of the sensitivity matrix. If, the correlation coefficient for any two parameters (e.g., p i and p j ) approaches unity, then these two parameters should have similar values, and vice versa. This technique of sensitivity-based correlation constraints has been demonstrated to alleviate numerical difficulties associated with ill-conditioning [2]. 6 th IWSHM, September 2007 5

Upon implementing all of the aforementioned considerations, and the introduction of a weighting matrix W to represent relative confidence in identified properties, the sequential minimization problem is expressed as min Δp WS (k ) Δp Wr (k ) L 2 2 such that 0 p i (k ) 1& p i (k ) p j (k ) R ij (7) the solution of which yields Δp at the k th iteration that is subsequently used for updating; p (k+1) = p (k) +Δp. The sequence continues by solving (7) with the updated parameters until the residual norm becomes less than a specified tolerance. To increase the effectiveness of the modal-based updating algorithm, an FRFbased technique may be included. In this case, the iterative form of error residual and sensitivity matrix are defined as, S T [ Re(S F ) Im(S F ) S M ], r T { Re(r F ) Im(r F ) r M } (8) where the subscripts F and M correspond to FRF- and modal-based formulations, respectively. The modal sensitivity and residual were described earlier; and the new FRF-based residual and sensitivity matrices are given by S F = B,1 (ω 1 ) % H(ω 1 ) L B, n (ω 1 ) % H(ω 1 ) M O M B,1 (ω m ) % H(ω m ) L B, n (ω m ) % H(ω m ), r F = l B(ω 1 ) % H(ω 1 ) M l B(ω m ) % H(ω m ) (9) where B(ω j ) and B,i (ω j ) denote the dynamic stiffness and its derivative with respect to the i th parameter evaluated at the j th sampled frequency ω j and H(ω) is the experimentally obtained transfer function. APPLICATIONS The initial FEM of the Factor building underestimates both the mass and stiffness of the actual building. Mass is underestimated because contributing live and dead loads were not considered. Stiffness is also underestimated because NSC contributions were omitted. A simple stick model with mass and stiffness properties directly related to the updating parameters is superposed onto the initial FEM to include these contributions. Each story of the stick model is assigned an effective EW, NS, and torsional stiffness, resulting in 45 stiffness parameters, and translational masses and mass moment of inertia based on a uniformly distributed weight resulting in 15 mass parameters. Defined as ratios ranging from 0 to 1, the properties assigned to the stick model are those multiplied by preassigned maximum expected physical quantities (displayed in the last row in the table in Figure 4). Modal properties of the updated model are displayed in TABLE I, and selected additional stiffness and mass quantities, based on updated parameters are displayed in Figure 4. The agreement between the analytical and identified modal properties is drastically improved as a result of model updating. Furthermore, the response prediction of the updated model is enhanced. 6 th IWSHM, September 2007 6

Figure 4. Factor building comparison of predicted (black) and measured (gray) roof accelerations where the E i and E u are the L 2 norms of the error between initial and updated models, alongside table of final values of select updated parameters. Figure 5. Four Seasons (a) comparison of measured and predicted transfer functions, (b) example cartoon of updated stiffness parameter distribution, and (c) table of select final effective stiffness factors based on updated parameters. In model-updating of the Four Seasons Building, the structural members were grouped into three sub-frames along each direction (i.e., West perimeter frame, East perimeter frame and NS-interior frame for NS stiffness, Figure 5b). Each sub-frame was sub-divided according to their member types and their vertical location. Flexural stiffness values in each group were selected as updating parameters. Shear and axial stiffnesses were set according to values recommended by FEMA 356 and were held fixed during model updating computations (i.e., they were not included in the updating parameter set) as they have minor effects on the structural system s dynamic response. Because the diaphragms are assumed to be rigid in-plane, the mass of each diaphragm can be expressed using only four parameters the translational mass, the radius of gyration, and distances of the center of mass from the reference point along x and y directions. Herein, we only select the translational and the rotational mass of each story as updating parameters, and exclude the locations of the centers of mass, as the building has a fairly symmetric plan. Finally, modal damping ratios of the 7th through the 15th modes 6 th IWSHM, September 2007 7

were selected as damping parameters, since the 1st to 6th modal damping ratios were already available from the system identification analysis and were believed to be reasonably accurate. As per the preceding assumptions, a total of 71 updating parameters (comprising 10 mass, 52 stiffness, and 9 damping parameters) were selected for updating. An appropriate set of absolute bounds was assigned to the updating parameters by considering their relative uncertainties. For example, narrower bounds were applied to the mass parameters (since more accurate estimations of mass properties were deemed possible using measured material weights). Figure 5a displays good comparison of updated and measured transfer functions and Figure 5c lists selected effective stiffness factors, based on updated parameters. Factors that were significantly reduced from their initial values of 0.5 are bold-italicized and correspond well to locations of observed damage. CONCLUSIONS The N4SID algorithm was used to identify properties of the first nine modes of the Factor building and the first six modes of the Fours Seasons building. FEMs of each structure were created and updated by using an improved FRF- and/or modalbased sensitivity method. The predicted and measured acceleration responses of the updated models agree quite well. Final parameters used in updating the Factor building FEM represent additional stiffnesses and mass that were not readily modeled. For the Four Seasons building, the predicted reduction in effective stiffness factors were in general agreement with observed damage patterns. Nevertheless, these results cannot be considered unique despite the dense sensor arrays, because they ultimately depend upon user-defined weights and constraints. ACKNOWLEDGEMENTS The authors are grateful to E. Yu, Y. Lei, D. Whang, T. Sabol, I. Stubailo, M. Kohler, S. Taylor-Lange, G. Farrar, M. Contreras, X. Du, W. Elmer, and A. Parker. This project was supported in-part by the Center for Embedded Networked Sensing under the NSF Cooperative Agreement CCR-0120778, and conducted in-part with equipment purchased and integrated into the nees@ucla Equipment Site with support from NSF Cooperative Agreement CMS-0086596. Additional support was provided by NSF award CMS-0301778 for testing and the analytical studies. REFERENCES 1. Skolnik D, Lei Y, Yu E, Wallace JW. 2006. Identification, Model Updating, and Response Prediction of a 15-Story Steel Frame Building, Earthquake Spectra, 22(3):781-802. 2. Yu E, Taciroglu E, Wallace JW. 2007. Parameter identification of framed structures using an improved finite element model updating method, Part I: Formulation and verification; Part II: Application to experimental data. Earthquake Eng. & Structural Dynamics, 36: 619-660. 3. Van Overschee P, De Moor B. 1994. N4SID: Subspace Algorithms for the Identification of Combined Deterministic-Stochastic Systems, Automatica, Special Issue on Statistical Signal Processing and Control, 30(1):75-93. 6 th IWSHM, September 2007 8