31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008

Damage Detection without Baseline Modal Parameters Utilizing the Baseline Stiffness Method and Independent Component Analysis for Modal Parameter Extraction Rodríguez-Rocha R. (1), Rivero-Angeles F.J. (2), Gomez-Ramírez E. (3) (1) Escuela Superior de Ingeniería y Arquitectura, IPN. Avenida Juan de Dios Batiz. Edificio 12, Col. Zacatenco, CP 07738, México DF, México, [email protected] (2) Universidad Justo Sierra, Escuela de Arquitectura, Eje Central Lázaro Cárdenas 1150, Col. Nueva Industrial Vallejo, CP 07700, México DF, México, [email protected] (3) LIDETEA. Posgrado e Investigación, Universidad La Salle. Benjamín Franklin 47, Col. Condesa, CP 06140, México DF, México, [email protected] Introduction Seismic instrumentation acquires multivariate data, which in turn, a suitable representation of this data is a common problem encountered in many disciplines such as structural dynamics, statistics, data analysis, signal processing and artificial neural networks. System identification has gained much interest by researchers for quite a few years now including a wide range of approaches (Kalouptsidis & Theodoridis, 1993); these methods include linear regressions, Wiener and Kalman filters, state-space predictors, adaptive and tracking methods, nonlinear regressions, least mean squares, recursive and non-recursive deterministic or stochastic approaches, adaptive filters, deconvolutions, spectral analysis, equalization, echo cancellation, neural networks and artificial intelligence, among others. In the damage detection area several methods to compute a reference state of structures without baseline modal information have been developed. Stubbs & Kim (1996) proposed the Sesitivity Method to compute baseline modal parameters from a structure iteratively. However, the algorithm may not converge depending on initial conditions. Kharrazi et al. (2000) applied sensitivity techniques to fit the analytical model and structure using experimental measurements. The Stiffness-Mass Ratios Method (Barroso & Rodríguez, 2004) determines the undamaged state of only shear beam buildings with regular mass distribution per floor using damaged information. This method does not identify damage per structural element, solely per story, which is a limitation. A recently developed linear method is Independent Component Analysis (ICA) (Hyvärinen, 1999), in which the output representation minimizes the statistical dependence of the components of the original representation and captures the essential structure of the data. In this paper, the Baseline Stiffness Method (BSM) is presented to assess damage in buildings without baseline modal parameters. The proposed method utilizes output responses from damaged structures and the approximate undamaged lateral stiffness of the first story to determine a pre-damage state. In order to extract modal parameters the ICA is applied to acceleration signals from instrumented structures. These identified parameters are used to adjust and compare stiffness quantities with the BSM in order to detect loss of stiffness on each element of the structure. The authors use ICA to develop a representation of the acceleration output acquired by a seismic instrumentation. Fourier transform of this new representation yields clearer spectral analysis to select the structural frequencies. Moreover, the recovered signals show one large peak associated to a particular frequency, avoiding pain staking peak selection in normal Frequency Response Functions. Finally, mode shapes are

387

31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008 obtained by conventional spectral analysis with transfer functions and phase angles of the original acceleration output. The structure of the paper is the following; first, the ICA and BSM methods are presented to extract modal parameters and detect damage without baseline modal information respectively. Then, two study cases from the literature are studied. The effect of limited modal information is also investigated. Results are discussed which demonstrate the feasibility of the methods. Finally, conclusions are stated showing advantages and limitations of the proposed methodology. Independent Component Analysis Method An important problem in structural engineering is to find a suitable transformation of the data to facilitate the analysis for subsequent processes, such as pattern recognition, visualization, system identification or damage detection. Consider the case of a building structure: the acceleration output at each story x corresponds to a realization of an m -dimensional discrete-time signal x ( t ) , t = 1, 2,… . Then the components si ( t ) are called source signals, which are usually original, uncorrupted signals or noise sources. Such sources are often statistically independent from each other, and thus, the signals could be considered as linear mixtures xi of a transformed signal (Hyvärinen, 1999). The previous paragraph hints to a very important issue: the acceleration output at each level could be considered as a linear mixture of independent sources (independent signals originated from the frequencies and mode shapes of the structure). Independent Component Analysis (ICA) is a recently developed linear transformation method which separates the sources from the acquired data. The observed m -dimensional random vector is denoted by x = ( x1 ,… , xm ) . ICA of the random vector x consists of finding a linear T

transform s = Wx so that the components si are as independent as possible, in the sense of

maximizing some function F = ( s1 ,… , sm ) that measures independence. In that sense, ICA of a random vector x consists of estimating the following generative model for the data x = As , where A is a constant m × n mixing matrix, and the latent components si in the vector

s = ( s1 ,… , sn ) are assumed independent (Jutten & Herault, 1991). T

In the previous model, the noise has been omitted since acceleration output usually contains noise during acquisition. The choice of the model is a tractable approximation of the more realistic noisy model, yet the results justify the use of the simpler model because it seems to work for certain kinds of real data. The model is asymptotically equivalent to the natural relation W = A−1 with n = m . A very simple MATLAB (The MathWorks, 2007) code is given in Parra (2007), though this method is not robust, with poor statistical and numerical performance; nonetheless, it could be a good start for tutorial purposes. The reader is referred to the works of Parra & Sajda (2003) and Cardoso & Souloumiac (1993) for better results and improved performance. Adaptive algorithms based on stochastic gradient descent may be problematic where no adaptation is needed. Convergence is often slow and depends of the learning rate sequence. A fixed-point algorithm, named FastICA was introduced using kurtosis or general contrast functions. The expectations are estimated using sample averages over a sufficiently large sample of the input data (Hyvärinen & Oja, 1997; Hyvärinen, 1999b). This algorithm is parallel and distributed, but is not adaptive. FastICA uses sample averages computed over larger samples of data. Hyvärinen (1999c) also showed that when FastICA is used with symmetric decorrelation, it is essentially equivalent to a Newton method for maximum likelihood estimation, that is, FastICA is a general algorithm that can be used to optimize contrast functions. Baseline Stiffness Method The Baseline Stiffness Method (BSM) is presented to detect damage in buildings without baseline modal parameters (undamaged state). This method utilizes stiffness-mass ratios to

388

31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008 determine a reference state (baseline) from the structure based on modal parameters from the damaged system and the approximated lateral stiffness from the first story. This identified reference state is compared to the damaged one. For a damaged plane frame of s number of floors and i mode shapes and performing signal processing techniques, natural frequencies ϖ and their corresponding mode shapes [φ ] can

[ ]

[ ]

be computed. Lateral stiffness and mass matrix, K and M respectively, are unknown and

of dimensions s × s . On the other hand, it is possible to compute a vector {u} of ratios ki mi (Barroso & Rodríguez, 2004) with dimensions 2s − 1× 1 :

⎧⎛ ⎞ {u} = ⎪⎨⎜⎜ k1 ⎟⎟ ⎪⎩⎝ m1 ⎠

⎛ k2 ⎞ ⎜⎜ ⎟⎟ ⎝ m1 ⎠

⎛k ⎞ ⎛ k2 ⎞ ⎜⎜ ⎟⎟ … ⎜⎜ i ⎟⎟ ⎝ m2 ⎠ ⎝ mi ⎠

⎛ k ⎞⎫⎪ ⎛ k i +1 ⎞ ⎟⎟ … ⎜⎜ s ⎟⎟⎬ ⎜⎜ ⎝ ms ⎠⎪⎭ ⎝ mi ⎠

T

(1)

This vector {u} is computed utilizing modal parameters from the damaged structure and the first story approximated lateral stiffness k1 assuming a shear beam behavior. It is well known this assumption is valid for limited real cases, however, this is proposed just as an initial condition and the flexural effect will be included later on. In this sense, k1 can be determined as: 12 EI1 (2) k1 = ∑ h13 Substituting k1 into equation (1), some parameters pi are obtained using back substitution as: p1 = k1

pi − j =

pi − ( j +1) u( j + 4) u( j + 5)

pi −1 =

ki =

pi − 2 u4 u5

for j = 2,3,...,( i − 2 )

(3)

pi −1u2 u3

[ ]

Once all ki are known, the lateral stiffness matrix of the structure without damage K can be determined. In order to calculate mi , m1 is utilized in equation (3) instead of using k1 . These

[ ]

mi are used to obtain the mass matrix of the structure M . The former approach was applied to buildings without shear beam behavior and it was observed that an approximated mass matrix Ma is obtained, which differs in magnitude to M . The difference is null if k1 is k1 c , where c is a coefficient that adjusts shear to flexural behavior and it was found to correspond

[ ]

[ ]

[ ][ ]−1 . Thus, when the adjustment by

to the greatest eigenvalue of M Ma

k1 c , for structures

[ ]

without shear beam behavior is performed, the BSM provides its undamaged state K . Simultaneously, a mathematical model of the structure is created considering connectivity and geometry of its structural elements and a unit elasticity modulus. Thus, approximated stiffness matrices [kai ] for each element are obtained. The global approximated stiffness matrix of the structure is

[Ka] = ∑ [kai ]

(4)

According to Escobar et al. (2005),

[ ]

[Ka ]

can be condensed to obtain [K a ] using the

transformation matrix T as:

389

31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008

[Ka] = [T ] [Ka][T ] T

where

(5)

⎤ [I ] ⎡[ Ka11 ] [ Ka12 ] ⎤ ⎥ ; [ Ka ] = ⎢ ⎥ −1 ⎣[ Ka21 ] [ Ka22 ]⎦ ⎣⎢ − [ Ka22 ] [ Ka21 ]⎦⎥ ⎡

[T ] = ⎢

(6)

For a shear beam building, [K ] and equation (5) just differ on material properties, specifically, on the magnitude of the elasticity modulus that can be represented using the matrix [P ] as K = [P ] K a . Solving [P ] from last equation yields:

[ ]

[ ]

[P] = [K ][K a]−1

(7) On the other hand, stiffness matrices for each structural element of the undamaged state of the structure are calculated as: [ki ] = P[kai ] (8) Where P is a scalar that adjusts the material properties of the structure from the proposed model. This scalar is obtained as the average of the eigenvalues of matrix [P ] , given in equation (7). Eigenvalue computations are performed because are useful to obtain characteristic scalar values of a matrix, in this case [P ] . It was found that the average of these eigenvalues is precisely P . Once the undamaged state of the structure, represented by [ki ] , is identified and condensed, it is compared against the stiffness matrix of the damaged

[ ]

structure K d using the Damage Submatrices Method (DSM, Rodríguez & Escobar, 2005). This method is applied to locate and determine magnitude of damage, in terms of loss of stiffness, in percentage, at every structural element. According to Baruch & Bar Itzhack

[ ]

(1978), K d can be computed from measured modal information. Thus, the condensed stiffness matrix of the damaged system can be reconstructed as:

[Kd ] = ⎡⎢⎣ [K ]− [M ][Z ]⎤⎥⎦[H ] + [M ][q][Ω]2 [q] [M ] T

[]

[

]

(9) 1 2 ; [φ ] is the modal

T [ ] [ ][ ]T [ ] [ ] [ ][ ]T where [H ] = [I ] − [Y ] ; Y = q q M ; Z = q q K ; [q] = [φ][φ] [M ][φ] 2 matrix of the structure and [Ω ] is a diagonal matrix containing the eigenvalues of the system.

Numerical Examples Modal parameter extraction of a four-story shear frame utilizing ICA In order to extract modal parameters, the ICA method was applied to a four-story shear building studied by Bernal & Gunes (2000), Figure 1. Mass and lateral stiffness values are k=7.5x107 N/m. m1=3600 kg, m2=m3=2850 kg, m4=1800 kg. m4 m3

k

m2

k

m1

k k

Figure 1. Four-story shear building (Bernal & Gunes, 2000) This model was excited with random normal-distributed noise at a sampling rate of 0.02 seconds. Figure 2 shows the Fourier spectra of the acceleration output signals. Note that the

390

31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008 peak selection below 0.04 seconds becomes quite difficult to extract. Figure 3 shows the Fourier spectra of the extracted sources from ICA. Amplitude 500 0

1000 0 2000 1000 0 2000 1000 0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Period (sec.)

Figure 2. Fourier spectra of the acceleration output at each floor. Amplitude 5000

0

1000 0

500 0 200 100 0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Period (sec.)

Figure 3. Fourier spectra of the unmixed sources, extracted with ICA. There is no physical interpretation as to which level does it correspond, instead, they are the unmixed original sources for each frequency. Note here that each signal contains only one large peak, quite easy to select, even for periods below 0.04 seconds. The extracted periods from Figure 3 are shown in Table 1. A comparison of these results and the ones from Bernal & Gunes (2000) is also presented. Table 1. Periods (sec.) using ICA applied to the four-story shear building Computed Error Theoretical by ICA (%) 0.103 0.104 -1.0 0.038 0.036 5.3 0.025 0.025 0.0 0.020 0.020 0.0 It can be noted from Table 1 that errors are less than 10%, acceptable in engineering.

391

31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008

Damage detection of a three-story frame utilizing BSM Figure 4 shows the frame building model proposed by Biggs (1964). According to Biggs, the flexural stiffness of the columns from story 1 is 2,688,218.5 N/m, stories 2 and 3 is 3’887,846.3 N/m. The weights for floors 1 to 3 are 241,537.7 N, 226,857.2 N, and 113,433.5 N, respectively. 9.1 m 9

Floor 2

2 @ 3.0 m

Floor 3

8 2

Floor 1 4.6 m

6

3

5 7 4

1

X Figure 4. Three-story frame (Biggs, 1964) Two damage cases, D1 and D2, were simulated. D1 consists of a 10% loss of stiffness on element 8. For D2, elements 2 and 4 were damaged 20% and 30% respectively. Again, ICA was used to extract the periods of the structure as shown in the previous section. To obtain mode shapes the conventional spectral analysis (Bendat & Piersol, 1986) was used. The transfer function of the power spectral density function (PSD) of each level to the ground PSD was calculated. Then, for each frequency obtained with ICA, the amplitude of the transfer function was calculated and divided by the amplitude of the transfer function of the ground. This method gives the amplitude of the mode shape. The polarity of the mode shape was computed with the phase angle of the cross PSD functions. If the angle is between 0 and 90º, a (+) sign is given, if it is between 90 and 180º, a (-) sign is assigned; thus, obtaining the complete mode shape. Modal parameters computed using ICA and the simulated ones are presented in Table 2.

Mode T (sec.) ICA Mode shape. ICA T (sec.) Simulated Error (%)

Table 2. Modal parameters for the three-story frame Undamaged case Damage case D1 Damage case D2 1 2 3 1 2 3 1 2 3 0.743

0.260

0.179

0.745

0.260

0.179

0.802

0.274

0.183

1.000 0.893 0.597

-1.000 -0.117 0.993

1.000 -0.699 0.390

1.000 0.893 0.598

-1.000 -0.117 0.993

1.000 -0.703 0.390

1.000 0.907 0.618

-1.000 -0.222 1.00

1.000 -0.948 0.298

0.744

0.260

0.179

0.744

0.260

0.179

0.803

0.274

0.183

0.2

0.0

0.0

-0.1

0.0

0.0

0.1

0.0

0.0

In Table 2, the period values (T) and mode shapes for the undamaged case calculated with ICA are very similar to those reported by Biggs (1964). Note that error values are 0.2% or smaller. The period T, and the related mode shape values for damage cases D1 and D2, computed using ICA, were utilized by the BSM to detect structural damage for both scenarios, knowing and not knowing the undamaged case (baseline modal parameters).

392

31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008 Figure 5 presents degradation of damage in percentage computed by the BSM. It can be observed from Figure 5 that the proposed damage identification method localized damaged elements for both cases. Note for D1, that when baseline modal parameters are not known the method determined a higher value of degradation of stiffness. For D2 some false damaged elements were identified, however, degradation values were smaller than the real ones. Loss of 30 stiffness (%)

Loss of 20 stiffness (%)

20 10 10 Number

0

0 1

2

3

4

5

6

7

8

1

9

a) Damage case D1 With baseline modal parameters

2

3

4

5

6

7

8

9 of element

b) Damage case D2 Without baseline modal parameters

Figure 5. Degradation of stiffness using the BSM applied to the three-story frame, with and without baseline modal parameters. The effect of limited modal information was also studied when baseline modal parameters are unknown. The number of identified mode shapes from the damaged structure computed was varied from 3 to 1. Figure 6 presents these results for damage case D2. Loss of stiffness (%) 30

20

10

0

Number 1

2

Three modes

3

4

5

6

Two modes

7

8

9 of element

One mode

Figure 6. Degradation of stiffness for Damage case D2 using the BSM applied to the threestory frame, without baseline modal parameters. As it was expected, the fewer the number of modes used to determine the damaged stiffness matrix, the less precise the method is. When all modes were utilized, error values were 2.6% and 9.6% for elements 2 and 4 respectively. Note also that when 2 modes were utilized, the BSM localized adequately the damaged elements. This is an advantage of the method since in practice only some modes can be extracted from dynamic measurements, not all of them. For the one mode case the BSM identified the most damaged element along with several false locations. Conclusions It has been proven that ICA is a powerful tool to obtain the structural periods and mode shapes. Also, for these particular study cases, the BSM with a sufficient number of mode shapes determined location and severity of damage in terms of loss of stiffness from a structure, without baseline references. When only a limited number of modes (1 or 2) were

393

31st General Assembly of the European Seismological Commission ESC 2008 Hersonissos, Crete, Greece, 7-12 September 2008 used to fit the damaged stiffness from dynamic measurements, the BSM was capable to determine the location of damage.

Acknowledgement The authors would like to gratefully acknowledge the Instituto Politécnico Nacional for supporting this research project, No. 20080145. References Barroso L. & Rodríguez R. (2004). Damage detection of a benchmark structure without baseline information, ASCE Journal of Engineering Mechanics, 130(2), 142-151. Baruch M. & Bar Itzhack IY. (1978). Optimal weighted orthogonalization of measured modes, American Institute of Aeronautics and Astronautics, 16(4), 346-351. Bendat J.S. & Piersol A.G. (1986). Random data. Analysis and measurement procedures, John Wiley & Sons. Bernal D. & Gunes B. (2000). Performance of an observer state-space identification in the presence of mild nonlinearities, 2000 American Control Conference, Chicago, Illinois, 2, 986-990. Biggs J.M. (1964). Introduction to structural dynamics, McGraw-Hill. Cardoso J.F. & Souloumiac A. (1993). Blind beamforming for non Gaussian signals, IEE Proceedings-F, 140(6), 362-370. Hyvärinen A. (1999). Survey on Independent Component Analysis, Neural Computing Surveys, 2, 94-128. Hyvärinen A. (1999b). Fast and robust fixed-point algorithms for independent component analysis, IEEE Transactions on Neural Networks, 10(3), 626–634. Hyvärinen A. (1999c). The fixed-point algorithm and maximum likelihood estimation for independent component analysis, Neural Processing Letters, 10(1), 1–5. Hyvärinen A. & Oja E. (1997). A fast fixed-point algorithm for independent component analysis, Neural Computation, 9(7), 1483-1492. Jutten C. & Herault J. (1991). Blind Separation Sources, Part I: An adaptive algorithm based on neuromimetic architecture, Signal Processing, 24, 1-10. Kalouptsidis N. & Theodoridis S. (1993). Adaptive System Identification and Signal Processing Algorithms, Prentice Hall International Series in Acoustics, Speech and Signal Processing. Kharrazi M.H.K., Ventura C.E., Brincker R. & Dascotte E. (2000). A study on damage detection using output-only modal data, 20th International Modal Analysis Conference, Los Angeles, California. Parra L. & Sajda P. (2003). Blind Source Separation via Generalized Eigenvalue Decomposition, Journal of Machine Learning Research, 4, 1261-1269. Parra L. (2007). Quickie Blind Source Separation. Revised on-line on 2007/02/11, http://newton.bme.columbia.edu/~lparra/publish/quickiebss.html Rodríguez R., Escobar J.A. & Gómez R. (2005). Detección de daño en edificios utilizando submatrices de daño, XV Congreso Nacional de Ingeniería Sísmica, Mexico City, 15,1-7. Stubbs N. & Kim J. (1996). Damage localization in structures without baseline modal parameters, American Institute of Aeronautics and Astronautics Journal, 34(8). The MathWorks, Inc. (2007). Matlab ® The Language of Technical Computing, R2007a.

394