Tracking modal parameters from building seismic response data using recursive subspace identification algorithm

Tracking modal parameters and estimating the current structural state of a building from seismic response measurements, particularly during strong earthquake excitations, can provide useful information for building safety assessment and the adaptive control of a structure. Therefore, online or recursive identification techniques need to be developed and implemented for building seismic response monitoring. This paper develops and examines different methods to track modal parameters from building seismic response data. The methods include recursive data‐driven subspace identification (RSI‐DATA) using Givens rotation algorithm, and RSI‐DATA using Bona fide algorithm. The question on how well the results of RSI‐DATA reflect the real condition is investigated and verified with a bilinear SDOF simulation study. Time‐varying modal parameters of a four‐story reinforced concrete school building are identified based on a series of earthquake excitations, including several seismic events, large and small. Discussions on the different methods' ability to track the time‐varying modal parameters are presented. The variation of the identified building modal frequencies and damping ratios from a series of event‐by‐event seismic responses, particularly before and after retrofitting of the building is also discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

Discussion of user-defined parameters for recursive subspace identification: Application to seismic response of building structures

Structural damage assessment under external loading, such as earthquake excitation, is an important issue in structural safety evaluation. In this regard, an appropriate data analysis and system identification technique is required to interpret the measured data and to identify the state of the structure. Generally, the recursive system identification algorithm is used. In this study, the recursive subspace identification (RSI) algorithm based on the matrix inversion lemma algorithm with oblique projection technique (RSI-Inversion-Oblique) is applied to investigate the time-varying dynamic characteristics. The user-defined parameters used in the RSI-Inversion-Oblique technique are carefully discussed, which include the size of the data Hankel matrix (i), model order to extract the physical modes, and forgetting factor (FF) to detect the time-varying system modal frequencies. Response data from the Northridge earthquake from the Sherman Oaks building (CSMIP) is used as an example to examine a systematic method to determine the suitable user-defined parameters in RSI. It is concluded that the number of rows in the data Hankel matrix significantly influences the identification of the time-varying fundamental modal frequency of the structure. An algorithmic model order selection method using the eigenvalue distribution of RSI-Inversion can detect the system modal frequencies at each appending data window without causing any abnormality.     

Identification of modal parameters of non‐classically damped linear structures under multi‐component earthquake loading

A method for parametric system identification of classically damped linear system in frequency domain is adopted and extended for non‐classically damped linear systems subjected up to six components of earthquake ground motions. This method is able to work in multi‐input/multi‐output (MIMO) case. The response of a two‐degree‐of‐freedom model with non‐classical damping, excited by one‐component earthquake ground motion, is simulated and used to verify the proposed system identification method in the single‐input/multi‐output case. Also, the records of a 10 storey real building during the Northridge earthquake is used to verify the proposed system identification method in the MIMO case. In this case, at first, a single‐input/multi‐output assumption is considered for the system and modal parameters are identified, then other components of earthquake ground motions are added, respectively, and the modal parameters are identified again. This procedure is repeated until all four components of earthquake ground motions which are measured at the base level of the building are included in the identification process. The results of identification of real building show that consideration of non‐classical damping and inclusion of the multi‐components effect of earthquake ground motions can improve the least‐squares match between the finite Fourier transforms of recorded and calculated acceleration responses. Copyright © 2006 John Wiley & Sons, Ltd.     

Two‐stage damage detection algorithms of structure using modal parameters identified from recursive subspace identification

Structural damage assessment under external loading, such as earthquake excitation, is an important issue in structural safety evaluation. In this regard, appropriate data analysis and feature extraction techniques are required to interpret the measured data and to identify the state of the structure and, if possible, to detect the damage. In this study, the recursive subspace identification with Bona‐fide LQ renewing algorithm (RSI‐BonaFide‐Oblique) incorporated with moving window technique is utilized to identify modal parameters such as natural frequencies, damping ratios, and mode shapes at each instant of time during the strong earthquake excitation. From which the least square stiffness method (LSSM) combined with the model updating technique, called efficient model correction method (EMCM), is used to estimate the first‐stage system stiffness matrix using the simplified model from the previously identified modal parameters (nominal model). In the second stage, 2 different damage assessment algorithms related to the nominal system stiffness matrix were derived. First, the model updating technique, called EMCM, is applied to correct the nominal model by the newly identified modal parameters during the strong motion. Second, the element damage index can be calculated using element damage index method (EDIM) to quantify the damage extent in each element. Verification of the proposed methods through the shaking table test data of 2 different types of structures and a building earthquake response data is demonstrated to specify its corresponding damage location, the time of occurrence during the excitation, and the percentage of stiffness reduction.     

Bayesian model selection for ARX models and its application to structural health monitoring

A Bayesian framework for model order selection of auto‐regressive exogenous (ARX) models is developed and applied to actual earthquake response data obtained by the structural health monitoring system of a high‐rise building. The model orders of ARX models are selected appropriately by the Bayesian framework, and differ significantly from the optimal order estimated by AIC; in fact, in many cases AIC does not even give an optimal order. A method is also proposed for consistently selecting the same ‘genuine’ modes of interest from the whole set of modes corresponding to each of the identified models from a sequence of earthquake records. In the identification analysis based on building response records from 43 earthquakes over 9 years, the modal parameters of the first four modes in each horizontal direction are estimated appropriately in all cases, showing that the developed methods are effective and robust. As the estimates of natural frequency depend significantly on the response amplitude, they are compensated by an empirical correction so that the influence of the response amplitude is removed. The compensated natural frequencies are much more stable over the nine‐year period studied, indicating that the building had no significant change in its global dynamic characteristics during this period. Copyright © 2010 John Wiley & Sons, Ltd.     

Structural identification using linear models and earthquake records

The problem of determining linear models of structures from seismic response data is investigated using ideas from the theory of system identification. The approach is to determine the optimal estimates of the model parameters by minimizing a selected measure-of-fit between the responses of the structure and the model. Because earthquake records are normally available from only a small number of locations in a structure, and because of noise in the records, it is necessary in practice to estimate parameters of the dominant modes in the records, rather than the stiffness and damping matrices of the linear model. A new algorithm is developed to determine the optimal estimates of the modal parameters. After tests with simulated data, the method is applied to a multi-storey building using records from the 1971 San Fernando earthquake in California. New information is obtained concerning the properties of the lower modes of the building and the time-varying character of the equivalent linear parameters.     

System identification of a building from multiple seismic records

This paper describes the identification of finite dimensional, linear, time‐invariant models of a 4‐story building in the state space representation using multiple data sets of earthquake response. The building, instrumented with 31 accelerometers, is located on the University of California, Irvine campus. Multiple data sets, recorded during the 2005 Yucaipa, 2005 San Clemente, 2008 Chino Hills and 2009 Inglewood earthquakes, are used for identification and validation. Considering the response of the building as the output and the ground motion as the input, the state space models that represent the underlying dynamics of the building in the discrete‐time domain corresponding to each data set are identified. The time‐domain Eigensystem Realization Algorithm with the Observer/Kalman filter identification procedure are adopted in this paper, and the modal parameters of the identified models are consistently determined by constructing stabilization diagrams. The four state space models identified demonstrate that the response of the building is amplitude dependent with the response frequency and damping, being dependent on the magnitude of ground excitation. The practical application of this finding is that the consistency of this building response to future earthquakes can be quickly assessed, within the range of ground excitations considered (0.005g–0.074g), for consistency with prior response—this assessment of consistent response is discussed and demonstrated with reference to the four earthquake events considered in this study. Inclusion of data sets relating to future earthquakes will enable the findings to be extended to a wider range of ground excitation magnitudes. Copyright © 2010 John Wiley & Sons, Ltd.     

Verification of system identification utilizing shaking table tests of a full‐scale 4‐story steel building

This paper verifies the feasibility of the proposed system identification methods by utilizing shaking table tests of a full‐scale four‐story steel building at E‐Defense in Japan. The natural frequencies, damping ratios and modal shapes are evaluated by single‐input‐four‐output ARX models. These modal parameters are prepared to identify the mass, damping and stiffness matrices when the objective structure is modelled as a four degrees of freedom (4DOF) linear shear building in each horizontal direction. The nonlinearity in stiffness is expressed as a Bouc–Wen hysteretic system when it is modelled as a 4DOF nonlinear shear building. The identified hysteretic curves of all stories are compared to the corresponding experimental results. The simple damage detection is implemented using single‐input‐single‐output ARX models, which require only two measurements in each horizontal direction. The modal parameters are equivalent‐linearly evaluated by the recursive Least Squares Method with a forgetting factor. When the structure is damaged, its natural frequencies decrease, and the corresponding damping ratios increase. The fluctuation of the identified modal properties is the indirect information for damage detection of the structure. Copyright © 2015 John Wiley & Sons, Ltd.     

Influence of lateral-load-resisting system on the earthquake response of structures—a system identification study

Analysis and comparison of the dynamic responses of three well instrumented (with accelerographs) high-rise buildings shaken during the 1984 Morgan Hill earthquake are presented. The buildings examined in the present work are (i) the Town Park Towers Apartment building, a 10-storey, concrete shear wall building; (ii) the Great Western Savings and Loan building, a 10-storey building with concrete frames and shear walls; and (iii) the Santa Clara County Office building, a 13-storey, moment-resistant steel frame building. The structures are located within 2 km of each other and, as may be confirmed by visual inspection of the recorded seismograms, experienced similar ground motions. One-dimensional and three-dimensional linear structural models are fitted to the observations using the modal minimization method' for structural identification, in order to determine optimal estimates of the parameters of the dominant modes of the buildings. The time-varying character of these parameters over the duration of the response is also investigated. Comparison of the recorded earthquake response of the structures reveals that the type of lateral-load-resisting system has an important effect on the dynamic behaviour of the structures because it controls the spacing of the characteristic modes on the frequency axis. The Santa Clara County Office building has closely spaced natural frequencies and exhibits strong torsional response and modal coupling. Its dynamic behaviour is contrasted with that of the Great Western Savings and Loan building which has well separated natural frequencies and exhibits small torsional response and no modal coupling. Strong modal coupling causes a beating-type phenomenon and makes earthquake response of structures different from that envisioned by codes.     

Identification of linear structural systems using earthquake-induced vibration data

This paper addresses the issue of system identification for linear structural systems using earthquake induced time histories of the structural response. The proposed methodology is based on the Eigensystem Realization Algorithm (ERA) and on the Observer/Kalman filter IDentification (OKID) approach to perform identification of structural systems using general input–output data via Markov parameters. The efficiency of the proposed technique is shown by numerical examples for the case of eight-storey building finite element models subjected to earthquake excitation and by the analysis of the data from the dynamic response of the Vincent-Thomas cable suspension bridge (Long Beach, CA) recorded during the Whittier and the Northridge earthquakes. The effects of noise in the measurements and of inadequate instrumentation are investigated. It is shown that the identified models show excellent agreement with the real systems in predicting the structural response time histories when subjected to earthquake-induced ground motion. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of the observed seismic response of a highway bridge

Strong-motion accelerograms obtained on the San Juan Bautista 156/101 Separation Bridge during the 6 August 1979 Coyote Lake, California, earthquake are used to examine the response of this multiple-span bridge to moderate levels of earthquake loading. Although the bridge was not damaged, the records are of significant engineering interest as they are the first to be recorded on a highway bridge structure in North America. A technique of system identification is used to determine optimal modal parameters for linear models which can closely replicate the observed time-domain seismic response of the bridge. Time variations in frequency and damping in the horizontal response are identified using a moving-window analysis. A three-dimensional finite element model is developed to study the bridge response in detail. The first two horizontal modal frequencies computed from this model are in excellent agreement with information obtained during the system identification analysis provided the finite element model's expansion joints are locked, preventing relative translational motions from occurring across the joints. Locking is confirmed by the observed seismic deformations of the structure in the fundamental mode. Fundamental vertical frequencies of the individual spans, predicted by the finite element model, are in very good agreement with ambient vibration test data.     

Finite element response sensitivity analysis of three-dimensional soil-foundation-structure interaction (SFSI) systems

The nonlinear finite element (FE) analysis has been widely used in the design and analysis of structural or geotechnical systems. The response sensitivities (or gradients) to the model parameters are of significant importance in these realistic engineering problems. However the sensitivity calculation has lagged behind, leaving a gap between advanced FE response analysis and other research hotspots using the response gradient. The response sensitivity analysis is crucial for any gradient-based algorithms, such as reliability analysis, system identification and structural optimization. Among various sensitivity analysis methods, the direct differential method (DDM) has advantages of computing efficiency and accuracy, providing an ideal tool for the response gradient calculation. This paper extended the DDM framework to realistic complicated soil-foundation-structure interaction (SFSI) models by developing the response gradients for various constraints, element and materials involved. The enhanced framework is applied to three-dimensional SFSI system prototypes for a pile-supported bridge pier and a pile-supported reinforced concrete building frame structure, subjected to earthquake loading conditions. The DDM results are verified by forward finite difference method (FFD). The relative importance (RI) of the various material parameters on the responses of SFSI system are investigated based on the DDM response sensitivity results. The FFD converges asymptotically toward the DDM results, demonstrating the advantages of DDM (e.g., accurate, efficient, insensitive to numerical noise). Furthermore, the RI and effects of the model parameters of structure, foundation and soil materials on the responses of SFSI systems are investigated by taking advantage of the sensitivity analysis results. The extension of DDM to SFSI systems greatly broaden the application areas of the d gradient-based algorithms, e.g. FE model updating and nonlinear system identification of complicated SFSI systems.     

Application of subspace identification technique to long‐term seismic response monitoring of structures

This paper addresses the issue of structural system identification using earthquake‐induced structural response. The proposed methodology is based on the subspace identification algorithm to perform identification of structural dynamic characteristics using input–output seismic response data. Incorporated with subspace identification algorithm, a scheme to remove spurious modes is also used to identify real system poles. The efficiency of the proposed method is shown by the analysis of all measurement data from all measurement directly. The recorded seismic response data of three structures (one 7‐story RC building, one midisolation building, and one isolated bridge), under Taiwan Strong Motion Instrumentation Program, are analyzed during the past 15 years. The results present the variation of the identified fundamental modal frequencies and damping ratios from all the recorded seismic events that these three structures had encountered during their service life. Seismic assessment of the structures from the identified system dynamic characteristics during the period of their service is discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Subspace system identification of support‐excited structures—part I: theory and black‐box system identification

This paper reviews the theoretical principles of subspace system identification as applied to the problem of estimating black‐box state‐space models of support‐excited structures (e.g., structures exposed to earthquakes). The work distinguishes itself from past studies by providing readers with a powerful geometric interpretation of subspace operations that relates directly to theoretical structural dynamics. To validate the performance of subspace system identification, a series of experiments are conducted on a multistory steel frame structure exposed to moderate seismic ground motions; structural response data is used off‐line to estimate black‐box state‐space models. Ground motions and structural response measurements are used by the subspace system identification method to derive a complete input–output state‐space model of the steel frame system. The modal parameters of the structure are extracted from the estimated input–output state‐space model. With the use of only structural response data, output‐only state‐space models of the system are also estimated by subspace system identification. The paper concludes with a comparison study of the modal parameters extracted from the input–output and output‐only state‐space models in order to quantify the uncertainties present in modal parameters extracted from output‐only models. Copyright © 2012 John Wiley & Sons, Ltd.     

Evolution of modal characteristics of a mid-rise cold-formed steel building during construction and earthquake testing

Vibration-based structural identification is an essential technique for assessing structural conditions by inferring information from the dynamic characteristics of structures. However, the robustness of such techniques in monitoring the progressive damage of real structures has been validated with only a handful of research efforts, largely due to the paucity of monitoring data recorded from damaged structures. In a recent experimental program, a mid-rise cold-formed steel building was constructed at full scale atop a large shake table and subsequently subjected to a unique multi-hazard scenario including earthquake, post-earthquake fire, and finally post-fire earthquake loading. Complementing the simulated hazard events, low-amplitude vibration tests, including ambient vibrations and white noise base excitation tests, were conducted throughout the construction and the test phases. Using the vibration data collected during the multi-hazard test program, this paper focuses on understanding the modal characteristics of the cold-formed steel building in correlation with the construction and the structural damage progressively induced by the simulated hazard events. The modal parameters of the building (i.e., natural frequencies, damping ratios, and mode shapes) are estimated using two input–output and two output-only time-domain system identification techniques. Agreement between the evolution of modal parameters and the observations of the progression of physical damage demonstrates the effectiveness of the vibration-based system identification techniques for structural condition monitoring and damage assessment.     

Stochastic subspace identification for output‐only modal analysis: application to super high‐rise tower under abnormal loading condition

The objective of this paper is to develop an online system parameter estimation technique from the response measurements through using the recursive covariance‐driven stochastic subspace identification (SSI‐COV) approach. In developing the recursive SSI‐COV, to avoid time‐consumption of singular value decomposition in recursive SSI, the extended instrumental variable version of the projection approximation subspace tracking method is used in SSI‐COV. Besides, to reduce the effect of noise on the results of identification, the preprocessing of data using recursive singular spectrum analysis technique is also presented to remove the noise contaminant measurements to enhance the stability of data analysis. On the basis of the proposed method, both the ambient vibration and seismic response data of a tower (Canton Tower) are used to observe the time‐varying system natural frequencies of a tower from its operating condition. Results from using off‐line SSI‐COV method under normal operating condition are also presented. Comparison on the identified time‐varying dynamic characteristics of the tower under normal operating condition and earthquake response of distanced earthquake event is discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Dynamic characteristics of a base isolated building from ambient vibration measurements and low level earthquake shaking

Ambient vibration tests were conducted on a base-isolated apartment building in Takamatsu, Japan, to determine the mode shapes and the associated natural frequencies and damping ratios at very low levels of excitation. The latest developments in signal analysis for modal decomposition are used to analyze the ambient response data. A finite element model of the building and isolators was calibrated and refined using the experimental results from the ambient vibration tests. This model was then used to simulate the recorded response of the building under excitation from a small earthquake. The finite element model, calibrated by ambient vibration data and the low level of earthquake shaking, provides the starting point for modelling the non-linear response of the building when subjected to strong shaking.     

System identification and modeling of a dynamically tested and gradually damaged 10‐story reinforced concrete building

This paper discusses the dynamic tests, system identification, and modeling of a 10‐story reinforced concrete building. Six infill walls were demolished in 3 stages during the tests to introduce damage. In each damage stage, dynamic tests were conducted by using an eccentric‐mass shaker. Accelerometers were installed to record the torsional and translational responses of the building to the induced excitation, as well as its ambient vibration. The modal properties in all damage states are identified using 2 operational modal analysis methods that can capture the effect of the wall demolition. The modal identification is facilitated by a finite element model of the building. In turn, the model is validated through the comparison of the numerically and experimentally obtained modal parameters. The validated model is used in a parametric study to estimate the influence of structural and nonstructural elements on the dynamic properties of the building and to assess the validity of commonly used empirical formulas found in building codes. Issues related to the applicability and feasibility of system identification on complex structures, as well as considerations for the development of accurate, yet efficient, finite element models are also discussed.     

Field measurements for calibration of simplified models of the stiffening effect of infill masonry walls in high-rise RC framed and shear-wall buildings

As a type of nonstructural component, infill walls play a significant role in the seismic behavior of high-rise buildings. However, the stiffness of the infill wall is generally either ignored or considered by simplified empirical criteria that lead to a period shortening. The difference can be greatly decreased by using a structural identification methodology. In this study, an ambient vibration test was performed on four on-site reinforced concrete high-rise buildings, and the design results were compared with the PKPM models using corresponding finite element(FE) models. A diagonal strut model was used to simulate the behavior of the infill wall, and the identified modal parameters measured from the on-site test were employed to calibrate the parameters of the diagonal strut in the FE models. The SAP2000 models with calibrated elastic modulus were used to evaluate the seismic response in the elastic state. Based on the load-displacement relationship of the infill wall, nonlinear dynamic analysis models were built in PERFORM-3 D and calibrated using the measured modal periods. The analysis results revealed that the structural performance under small/large earthquake records were both strengthened by infill walls, and the contribution of infill walls should be considered for better accuracy in the design process.     

System identification applied to long‐span cable‐supported bridges using seismic records

This paper presents the application of system identification (SI) to long‐span cable‐supported bridges using seismic records. The SI method is based on the System Realization using Information Matrix (SRIM) that utilizes correlations between base motions and bridge accelerations to identify coefficient matrices of a state‐space model. Numerical simulations using a benchmark cable‐stayed bridge demonstrate the advantages of this method in dealing with multiple‐input multiple‐output (MIMO) data from relatively short seismic records. Important issues related to the effects of sensor arrangement, measurement noise, input inclusion, and the types of input with respect to identification results are also investigated. The method is applied to identify modal parameters of the Yokohama Bay Bridge, Rainbow Bridge, and Tsurumi Fairway Bridge using the records from the 2004 Chuetsu‐Niigata earthquake. Comparison of modal parameters with the results of ambient vibration tests, forced vibration tests, and analytical models are presented together with discussions regarding the effects of earthquake excitation amplitude on global and local structural modes. Copyright © 2007 John Wiley & Sons, Ltd.