Experimental Mechanics (2008) 48:521–528 DOI 10.1007/s11340-008-9124-6
Identification and Damage Detection in Structures Subjected to Base Excitation G. Fraraccio & A. Brügger & R. Betti
Received: 24 July 2007 / Accepted: 15 January 2008 / Published online: 1 February 2008 # Society for Experimental Mechanics 2008
Abstract This paper presents an experimental study of different algorithms for the health monitoring of frame structures subjected to base excitation (e.g. earthquake ground motion). These algorithms use only the acceleration time histories of the input and of the response output and are tested for the identification of the dynamic characteristics of the structure (natural frequencies and damping ratios) and for detecting and quantifying any possible structural damage that occurs in the frame. Three algorithms were considered: (1) a frequency domain decomposition algorithm, (2) a time domain Eigensystem Realization Algorithm together with Observer Kalman Identification algorithm, and (3) a subsequent physical parameter identification algorithm (MLK). Through extensive experimental testing of a four-story steel frame model on a uniaxial shake table, the performance of the various methods as well as the inherent complications of physical instrumentation and testing are explored.
G. Fraraccio (*) Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma La Sapienza, Via Eudossiana, 18, 00184 Rome, Italy e-mail:
[email protected] A. Brügger : R. Betti Department of Civil Engineering and Engineering Mechanics, Columbia University, 610 S. W. Mudd Building, 500 West 120th Street, New York, NY 10027, USA A. Brügger e-mail:
[email protected] R. Betti e-mail:
[email protected]
Keywords System identification . Damage detection . Health monitoring . ERA . OKID . FDD . Experimental . Shake table
Introduction In recent years, there has been a growing interest in the civil engineering community on methodologies that are capable of detecting and quantifying structural damage that occurs in areas of buildings and bridges that are not easily accessible to engineers. Non-destructive techniques have been successfully developed for quantifying structural damage on single elements. Much work has been done over the past two decades in the field of crack detection in beams and plates from vibration data (e.g. Sundermeyer and Weaver [1], Hjelmstad and Shin [2], Kim et al. [3]). However, techniques and methodologies that have a “global” perspective in damage identification, focusing on the overall structure rather than on a single element, are still in the development phase, trying to overcome limitations imposed by computational and financial constraints and to define their range of applicability. Pivotal in all these global approaches to damage detection is the reliability of the structural model that can be identified. In civil engineering applications, the classical approach to dynamic system identification relies on the identification of the systems’ modal parameters (natural frequencies and damping ratios) that are subsequently used to update an approximate initial finite element model (FEM) of the system (e.g. Ewins [4], Mottershead and Friswell [5]). Because of the nature of the identified system’s parameters, many initial identification algorithms were developed in the frequency domain, taking advantage of the properties of the fast Fourier transform (FFT). One technique that has been recently
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developed in the realm of frequency domain is the frequency domain decomposition (FDD), proposed by Brincker et al. [6]. This method uses only information about the structural output and is considered to yield correct results when the unknown input is white noise and the system is lightly damped. While the assumption of low damping can be realistic for buildings and bridges (damping ratios in the order of 1–5%), the input excitation that usually acts on such structures, e.g. earthquake, wind, and traffic, are bandlimited, affecting the accuracy of the identified parameters. Recently, there has been a tremendous increase in transferring time domain identification methodologies, initially developed for mechanical and aerospace applications, to the analysis of buildings and bridges. In this transfer, however, the focus of the application shifted from the design of control systems to damage detection. Among these time domain identification algorithms, those that are gaining popularity in the civil engineering community are the Eigensystem Realization Algorithm (ERA; Juang and Pappa [7]) and its variations, among which the Observer Kalman filter Identification (OKID) algorithm shows great potential [8]. These algorithms use the time histories of the input and of the structural output to determine a minimal order state– space representation of the system from which it is then possible to extract an estimate of the natural frequencies and damping ratios. Also in this case, however, the presence of measurement noise and the band-limited input signal affect the outcome of the identification. One interesting consequence of these time domain ERA based approaches is that, under certain conditions, the identified system’s realization can be transformed into physical parameters of the structures, such as the stiffness, mass and damping matrices (Alvin [9], De Angelis et al. [10]). Identifying an accurate physical matrix like the stiffness matrix of a structure would be quite beneficial for damage detection: in fact, by comparing the stiffness matrices obtained for a benchmark state and for a damaged state, it is possible to extract valuable information about the damage location and, in some cases, about the amount of structural damage. In this paper, the results of an experimental study on the identification of the structural parameters and on the damage assessment using various identification algorithms are presented. The structure represents a scaled model of a four-story steel frame subjected to the various types of base excitation (earthquake recorded ground motions and white noise). The FDD and ERA/OKID algorithms are used to identify the natural frequencies and damping ratios while the MLK algorithm (De Angelis et al. [10]) is applied in the determination of the stiffness matrix for detecting structural damage. Two different damage scenarios are considered. The goal of this investigation is to experimentally quantify the sensitivity of these algorithms to the presence of
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structural damage and to test their efficiency in locating said damage within a reduced-order model.
Algorithm Description FDD Algorithm The FDD is a derivative of classical frequency domain peak-picking techniques developed to isolate closely spaced modes from output-only acceleration data (Brincker et al. [6]). By using the FFT of the structural outputs, the algorithm determines the output power spectral density (PSD) function, Gyy(wi), which is then decomposed at discrete frequencies through a singular value decomposition: Gyy ðwi Þ ¼ V i SViT
ð1Þ
where Si ¼ diag½ si1 si2 ::: sim contains the singular values and Vi ¼ ½ vi1 vi2 ::: vim contains the respective singular vectors. When the frequency wi is close to a structural mode, the first singular value becomes dominant and reaches a local maximum at wa, and the respective singular vector a is assumed to be an accurate estimate of the real mode shape [6]. Consequently all the singular values sij in the spectrum with a singular vector νij that is sufficiently parallel to the chosen mode shape are taken as part of the PSD of the single degree of freedom (SDOF), isolating the spectral power of the mode in question. The SDOF PSDs are then transformed back to the time domain by inverse FFT, and the resulting free decay time histories are used to obtain the modal frequencies and damping ratios. The SDOF decomposition yields exact results if (1) the unknown input is white noise, (2) the system is lightly damped and (3) different mode shapes are geometrically orthogonal. For civil structures subjected to earthquake and wind excitation, the assumption of light damping is acceptable while the input excitation is characterized by a narrow-banded spectrum. The algorithm may fail to properly isolate modes during peak-picking if the measured modes shape vectors are not orthogonal. If two geometrically different structural modes provide similar measured mode shapes, the algorithm cannot distinguish between the two singular vectors and fail to isolate the two modes from each other. ERA/OKID Algorithm The Eigensystem Realization Algorithm (Juang and Pappa [7]) estimates a minimal realization of the system in the form: zðt Þ ¼ Azðt Þ þ Buðt Þ yðt Þ ¼ Czðt Þ þ Duðt Þ
ð2a; bÞ
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in which the vector of the measured structural outputs, y(t), is related to the state vector, z(t), and to the input vector, u(t), through the system’s matrix A, the input matrix B, and the output matrices C and D. The last two matrices depend on the types of measurement available (displacement, velocities, or accelerations). The system’s matrix as well as the input and output matrices can be obtained from the singular value decomposition of the Hankel matrix containing the system’s Markov parameters. However, for lightly damped systems, the number of the system’s Markov parameters needed to provide a reliable representation of the system becomes too large and this introduces substantial numerical error that impairs a correct estimation of the system’s matrices. For this purpose, a significant improvement of the ERA methodology, in terms of both computation as well as accuracy, is represented by the OKID (Juang et al. [8]): in this approach, the Markov parameters of a fictitious observer are first determined and then used to retrieve the system's Markov parameters. Such parameters are then included in the Hankel matrix and used to obtain a first-order system representation. MLK Identification Algorithm One of the advantages of having identified a first-order state space model of the structure is that it can be used as the starting point to derive, through a set of transformations, the corresponding mass, damping and stiffness matrices. One of these methodologies (MLK formulation: De Angelis et al. [10]) determines such matrices using the complex eigenvalues and corresponding complex eigenvectors of the associated second-order damped eigenvalue problem. These complex eigenvalues and eigenvectors can be uniquely determined by applying proper transformations to the identified state, input, and output matrices of the state space model. By storing the complex eigenvalues and eigenvectors in the matrices Λ and y, respectively, the mass, damping and stiffness matrices of the structure can be expressed [10]: 1 M ¼ y Λ yT 1 K ¼ y Λ1 y T L ¼ M y Λ2 y T M
ð3a; b; cÞ
If the forcing term u(t) acting on the system is represented by an known arbitrary force applied to any arbitrary degree of freedom, then all three matrices can be easily obtained. However, for discrete systems, in the case of imposed support displacement, the transformation matrix correlating the different representation of the system cannot be uniquely determined, unless the value of at least one discrete mass is known.
Experimental Approach Experimental Setup The structural model (Fig. 1) is a four-story A36 steel frame with an interstory height of 533 mm and floor plate dimensions of 610×457×12.7 mm. The floors are braced diagonally in only one direction with the weak direction of bending facing the direction of excitation. The columns have cross-sectional dimensions of 50.8×9.5 mm while the diagonal braces measure 50.8×6.4 mm. All structural connections are bolted, allowing fast exchange and modification of structural members. Since the tightness of the structural bolts is critical for a precise identification of the structure’s stiffness, bolts were tightened after every table run to guarantee the precision of the recorded data. The structural model is mounted on an ANCO uniaxial hydraulic shake table, with a 1.5 by 1.5 m platform, capable of providing excitation in the frequency range of direct current to 150 Hz at peak acceleration of 3g. The model is fully instrumented with piezoelectric accelerometers, with seven channels of acceleration response on the structural model and one reference channel on the shake table’s platform. The sensors are attached magnetically to the column connection plates at each floor, at the height of the respective floor plate. The data acquisition scheme is outlined in Table 1. The sensor locations have been determined so to capture any threedimensional behaviour of the structure, induced either directly by the excitation or by the asymmetric damage. All dynamic data is acquired at 256 Hz.
Fig. 1 Model with accelerometer positions shown by arrows and damaged column location circled
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Table 1 Data acquisition scheme CH
Sensor Position
Orientation
1 2 3 4 5 6 7 8
Table reference 1st floor 2nd floor 3rd floor 4th floor 3rd floor 4th floor 4th floor
Weak Weak Weak Weak Weak Strong Strong Weak
Simulation of Damage Damage was simulated by reducing the cross-section of one column between the 2nd and 3rd floor (Fig. 1). Two damage cases were investigated: (1) damage case 1 (DC1) provides a 66% area reduction of the column, effecting a 22.2% stiffness reduction between these floors in the weak bending direction, and (2) damage case 2 (DC2) provides a 33% reduction in area, inducing a 13.9% reduction in stiffness in the weak direction. Hereafter, the undamaged case will be referred to as DC0.
(1994), and Kobe (1995)) and to a 30 s time history of band-limited white noise (frequency range of 1–80 Hz). In order to minimize errors due to uncontrolled variables, every test was repeated three times. To ensure that the structure was excited by the proper range of the time histories’ power spectra, a time scaling factor of 1 pffiffi3ffi was introduced and, to prevent yielding and additional unexpected damage, the input time histories were properly scaled in magnitude. Since the shake table control system invariably adds noise to the input time histories, the acceleration time histories measured by the table reference accelerometer were used in the identification algorithm. As shown in Fig. 2, the shake table’s response acceleration FFT shows a significant amount of high-frequency noise, with significant tight-banded 60 Hz interference in all excitation time histories, a common problem with control systems operating on alternating current. The source of the majority of such interference was traced back to the displacement controller of the shake table. However, such a very tight-banded interference did not significantly alter the structural behaviour since no major structural mode was present in the vicinity of 60 Hz.
Numerical Model Time History Inputs As external excitation, the model was subjected to base excitation in the form of earthquake acceleration time histories (El Centro (1940), Hachinohe (1983), Northridge Fig. 2 FFT of Reference and Actual shake table acceleration response time histories (Northridge earthquake)
To serve as a reference for the validation of the proposed approaches, a full-order FEM model with 48 degrees of freedom (df) was developed and analyzed. Because of the geometry of the frame, it is reasonable to assume that the FFT of Northridge Earthquake
45 Reference Table Response
40
35
30
25
20
15
10
5
0
0
10
20
30
40 Frequency (Hz)
50
60
70
80
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model behaves like a shear-type structure with masses lumped at each floor: this allows a model reduction from 48 to 12 df. Furthermore, the structure is excited only in the weak direction of motion, which for the undamaged case causes no measureable torsion and no transverse motion. Therefore, a reduced-order model with four df in the direction of excitation is considered in the identification of the four bending modes in the weak direction. For the identification of the benchmark structure, special attention will be given to the four weak bending modes. For each column, stiffness parameters were evaluated using ki ¼ 12EI L3 , where E is the Young’s modulus of structural steel, I is the column’s second moment of area in the direction of excitation, and L is the floor height. The physical parameters of stiffness and mass are shown in Table 2, with the corresponding undamped natural frequencies as w1 =3.75 Hz, w2 =10.78 Hz, w3 =16.53 Hz, w4 =20.27 Hz. The damping ratios were assumed to be 5%, 3%, 1%, and 1% for the first, second, third, and fourth modes, respectively. This structural model was excited by the El Centro earthquake acceleration time history with a peak ground acceleration of 0.20g and the acceleration response at each floor was computed at 256 Hz. These input and output time histories were then used to test the validity of the identification algorithms considered in this study. For the frequency domain decomposition identification, a Modal Assurance Criterion value of 0.9 was chosen to properly isolate the spectral power of each peak in the dominant singular values of the output PSD. The identified frequencies (w1 =3.85 Hz, w2 =10.94 Hz, w3 =16.60 Hz, w4 =20.22 Hz) are relatively accurate when compared to the theoretical ones, with errors ranging from 2.67% for the first natural frequency to 0.25% for the fourth frequency. On the contrary, the identified damping ratios show significant error, underestimating the actual modal damping of the system up to 37%. When using the ERA/OKID algorithm to identify the numerical model, the identified natural frequencies and damping ratios are quite accurate: for the natural frequencies, the maximum error of identification was a mere 0.012‰ while, for the damping ratios, the maximum error was 0.33%. Using the modal parameters and complex mode shapes from the OKID identification, the stiffness
matrix of the structural model was identified, assuming that (1) the diagonal mass matrix was known (Table 3), and (2) there is a 15% mass estimation error (Table 3). This second case was introduced to test the sensitivity of the identification algorithm to errors in estimating the structural mass. From these results, it appears that the stiffness matrix is identified quite accurately and that the algorithm is pretty stable with respect to the mass estimation (12.8% error in the stiffness for a 15% error in the mass estimation).
Damage Identification In order to provide a dependable damage identification scheme, a structure must first be identified in the undamaged or benchmark state. Once damage occurs, a deviation in the dynamic behavior of the structure with respect to the benchmark allows a damage identification algorithm to identify the presence and, depending on the power of the algorithm and the availability of data on the structure, the location of damage. For damage identification, the structural model was tested on the shake table using the previously listed excitation cases. Acceleration time histories for both the input excitation (table reference) and structural response (accelerometer nodes on structure) were recorded and processed by the FDD and the OKID/MLK algorithms. For the undamaged case, the FDD algorithm provides a very good identification of the structure’s natural frequencies. Four modes are clearly identified (Fig. 3). The identified frequencies are quite stable with respect to both the magnitude and spectral content of the input excitation (w1 =3.69 Hz±0.05 Hz, w2 =10.62±0.10 Hz, w3 =18.25± 0.08 Hz, w4 = 25.71 ± 0.19 Hz). Unlike the identified frequencies, the identified modal damping ratios show a strong dependence on the type of input time history used to excite the structure (ζ1 =0.99±0.11%, ζ2 =0.38±0.07%, ζ3 =0.30±0.04%, ζ4 =0.13±0.03%), highlighting a trend that indicates that a low upper frequency bound overestimates the damping estimation while a broad-banded excitation with spectral power in higher frequencies causes the damping to be underestimated. Similar results were obtained using the ERA/OKID algorithm using the order of the minimal realization fixed at the first four singular value complex
Table 2 Exact stiffness matrix and mass matrix of numerical model Stiffness 105 (N/m) 3.40 –1.70 0.00 0.00
Mass (kg) –1.70 3.40 –1.70 0.00
0.00 –1.70 3.40 –1.70
0.00 0.00 –1.70 1.70
37.00 0.00 0.00 0.00
0.00 37.00 0.00 0.00
0.00 0.00 37.00 0.00
0.00 0.00 0.00 37.00
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Table 3 Identified stiffness by OKID with exact masses (a) and 15% mass error (b) Exact Masses-Stiffness 105 (N/m) 3.40 -1.70 0.00 0.00 (a)
-1.70 3.40 -1.70 0.00
15% Mass Error - Stiffness 105 (N/m) 0.00 -1.70 3.40 -1.70
0.00 0.00 -1.70 1.70
3.91 -1.95 0.00 0.00 (b)
conjugate pairs (w1 =3.68 Hz±0.03 Hz, w2 =10.60±0.09 Hz, w3 =18.21±0.07 Hz, w4 =25.08±0.12 Hz and ζ1 =0.95± 0.19%, ζ2 =0.51±0.11%, ζ3 =0.47±0.1%, ζ4 =0.61±0.17%). Estimating that the mass of each floor is 37 kg, the stiffness matrix of the frame structure can be determined using the first-order realization provided by ERA/OKID. For the undamaged case, the identified stiffness matrix is presented in Table 4: although not purely tri-diagonal, the elements on the three central diagonals are predominant with respect to the remaining terms. Once the structure is identified in the benchmark phase, damage is introduced and the model is subjected again to the acceleration time histories. In both damage scenarios (DC1 and DC2), when identifying the structure by FDD, a fifth peak clearly appears near 24 Hz in the dominant singular value plot, Fig. 3, indicating the appearance of a fifth (torsional) mode. In the undamaged structure, the fourth mode was a coupled bending-torsional mode while now the introduction of damage in the structure has decoupled this mode. For the case of large damage Fig. 3 Dominant singular value for output PSD (FDD) for undamaged (DC0) and damaged (DC1) cases
-1.95 3.91 -1.95 0.00
0.00 -1.95 3.91 -1.95
0.00 0.00 -1.95 1.95
(DC1), the FDD algorithm provides stable results for the identified frequencies (w1 =3.53±0.02 Hz, w2 =10.24± 0.04 Hz, w3 =17.58±0.04 Hz, w4 =24.15±0.05 Hz, w5 = 25.90±0.13 Hz) while the identified modal damping ratios again depend strongly on the type of input excitation (ζ1 =0.84±0.14%, ζ2 =0.44±0.09%, ζ3 =0.26±0.04%, ζ4 = 0.20±0.03%, ζ5 =0.17±0.04%). Similar results are obtained for the DC2. Using the ERA/OKID algorithm, the identified natural frequencies are basically identical to those identified with FDD while the identified damping ratios show a different trend with respect to those from FDD. In fact, while the results from FDD show an increase in damping of the higher order modes and a decrease of the lower order modes when damage is introduced, the ERA/OKID results show a general decrease in damping ratios for all modes. However, this trend is very weak, especially in the lower order modes. Having the minimal realization from ERA/OKID corresponding to the two damage scenarios allows us to identify the corresponding stiffness matrices and, by Dominant Singular Values Plot DC0 DC1
5
10
4
10
3
10
2
10
1
10
0
10
0
3
6
9
12
15 18 Frequency (Hz)
21
24
27
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Table 4 Identified stiffness matrix for DC0 by MLK 5
DC0-Stiffness 10 (N/m) 5.94 -3.43 0.83 -0.14
-3.43 5.02 -2.75 0.40
0.83 -2.75 4.21 -2.03
-0.14 0.40 -2.03 1.67
comparing them with the initial “undamaged” one, to highlight the changes and quantify damage. In both damage cases, the stiffness matrix showed significant local deviation from the benchmark structure, as shown in Table 5. The identified stiffness matrices show a clear concentration of stiffness reduction in the term related to the interstory stiffness between the second and third floors of the structure. The identified stiffness reductions from the experimental data is 20.58% and 7.45% for DC1 and DC2, respectively, which correlate quite well with the theoretical stiffness reductions obtained from the four df FEM model (22.2% for DC1 and 13.9% for DC2). This discrepancy between the experimental and theoretical stiffness could be due to the fact that, in the determination of the theoretical interstory stiffness, the floor plates were assumed to be completely rigid and the diagonal bracing was assumed to provide no bending stiffness. It is important to mention that the bracing members have a stiffness of the same order of magnitude as the columns; although they are attached to the structure with a single plate connection (not with a double plate bolted connection as for the columns). If the bolts are sufficiently tight, partial fixity at the brace connections may effect a higher interstory stiffness since the bracing elements would be activated in bending. In order to verify the magnitude of this effect, a FEM analysis on the full-order model (48 df) was conducted with a symmetric set of virtual forces on the top floor in the weak direction. As expected, this analysis showed that the system with pinned–pinned connections on the bracing presents larger deformations (6.49%) than the system with fixed– fixed bracing. Therefore, the presence of bracing elements and their structural connections will alter the aggregate interstory stiffness of the theoretical model and, consequently, induce a reduction of the interstory stiffness change due to
damage in one column, bringing it closer to the identified experimental value. In order to define a minimum threshold of damage detection, the coefficient of variation with respect to input type of each stiffness coefficient, ki, being monitored for damage was considered. For the undamaged case, the greatest variation of any stiffness coefficient was 1.79% while DC1 had a maximum variation of 4.66% and DC2 1.24%. Therefore, even stiffness reductions smaller than 10% could be reliably identified with this algorithm.
Conclusions In this paper, three identification algorithms, the frequency domain decomposition, the Observer Kalman filter Identification, and the MLK algorithm, have been applied in the identification and damage detection for a four-story steel building model. Two different damage scenarios, corresponding to a stiffness reduction between second and third floor, were considered. Both the FDD and the ERA/ OKID algorithms provide a reliable identification of the natural frequencies of the structure. However, such parameters are not sufficiently responsive to damage and cannot be used as a practical indicator for structural damage. Furthermore, both algorithms failed to identify the structure’s damping to a sufficient degree of accuracy, showing that the identified damping coefficients were closely coupled to the spectral content of the input excitation. However, using the minimal realizations identified by ERA/OKID, it is possible to derive, through the MLK algorithm, an accurate estimate of the stiffness matrix of the structure which provides a reliable parameter for locating and quantifying structural damage. By comparing the identified stiffness matrices for the undamaged and the damage case, the stiffness reductions are indicative of the amount damage and its location. The identified stiffness reductions from the experimental data was estimated to be 20.58% and 7.45% for the two damage cases, respectively, values that match quite accurately the values obtained through a preliminary numerical analysis. The results of the ERA/OKID algorithm have proven to be robust against both input excitation magnitude and spectral content as illustrated by the low variances in the
Table 5 Identified stiffness matrix for DC1 (a) and DC2 (b) by MLK DC1-Stiffness 105 (N/m) 5.75 -3.05 0.61 -0.21 (a)
-3.05 4.18 -2.18 0.38
DC2-Stiffness 105 (N/m) 0.61 -2.18 3.60 -1.83
-0.21 0.38 -1.83 1.56
5.99 -3.28 0.81 -0.25 (b)
-3.28 4.63 -2.55 0.44
0.81 -2.55 4.00 -1.97
-0.25 0.44 -1.97 1.65
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physical parameter identifications obtained from the four different earthquake excitations as well as band-limited white noise. Three-dimensional effects, as often observed in real buildings, were also taken into account and internalized in the routine with the identification of the torsional mode in the damaged structure. Based on these results the ERA/ OKID algorithm boasts significant potential for on-line system identification and damage detection in structures subjected to earthquake excitation.
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