Proceedings of the IMAC-XXVII February 9-12, 2009 Orlando, Florida USA ©2009 Society for Experimental Mechanics Inc.

Limited Modal Information and Noise Effect on Damage Detection without Baseline Modal Parameters

Ramses Rodriguez-Rocha Associate Professor, Graduate School of Civil Engineering Escuela Superior de Ingeniería y Arquitectura, IPN Avenida Juan de Dios Batiz, Edificio 12, Col. Zacatenco, CP 07738, México, D. F. Tel. 5729-6000 Ext. 53087, [email protected] Francisco J. Rivero-Angeles Associate Professor, School of Architecture Universidad Justo Sierra Eje Central Lázaro Cárdenas 1150, Col. Nueva Industrial Vallejo, CP 07700, México, D. F. Tel. 5148-2382, [email protected] Eduardo Gomez-Ramirez Research and Advanced Technology Development Laboratory Graduate School of Engineering, Universidad La Salle Benjamin Franklin 47, Col. Hipódromo Condesa, CP 06140, México, D. F. Tel. 5278-9500 Ext. 2391, [email protected] NOMENCLATURE x Number of story;

m Length of signal vector

x(t)Discrete-time;

si  t  Source signal

xi Linear mixture of transformed signals;

F Maximazed function

A Constant mixing matrix;

s Number of floors

i Number of mode shapes;

 Natural frequency

  Mode shape matrix;

K  Lateral stiffness matrix

M  Mass matrix;

Ma Approximated mass matrix

ki mi Stiffness-mass ratio;

u Vector of ratios

k1 First approximated lateral stiffness;

pi Parameters

c Coefficient of flexural adjusment;

kai  Approximated stiffness matrix for element

Ka  Global approximated stiffness matrix;

T  Transformation matrix

P Material properties matrix;

P Material properties scalar

Kd Stiffness matrix of the damaged structure;

[H], [Y], [Z] Matrices from Baruch et al.

[q] Modal matrix;

Ω 2 Eigenvalue matrix

ˆMode shape matrix perturbed by noise;

N Level of noise

R Normally distributed random number

ABSTRACT In this paper, the Baseline Stiffness Method (BSM) is presented to identify location and severity of damage in buildings without modal parameters from the undamaged state. The proposed method utilizes solely dynamic responses from the damaged structure and the approximate undamaged lateral stiffness of the first storey to determine a baseline model. The Independent Component Analysis method is applied to acceleration signals from instrumented damaged structures to extract modal parameters. These identified parameters are used to adjust and compare stiffness quantities with the BSM utilizing eigenvalue computations in order to detect degradation of stiffness on structural elements. This study is focused on the effect of limited modal information and level of noise on the proposed BSM for damage detection. A numerical study case from the literature is studied. Results are discussed to demonstrate the applicability of the proposed method.

INTRODUCTION A common problem encountered in many disciplines such as structural dynamics, statistics, seismic instrumentation, data analysis, signal processing and artificial neural networks is to find a suitable representation of multivariate data. System identification has been amply studied by researchers including a wide range of approaches [1]; 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. The identified parameters are required for damage detection process. This damage detection area, conventionally, compares the undamaged and the damaged state. However, baseline modal information, very often, is not available since the structure was not instrumented prior damage. This is a very common problem in Mexico City. For this reason, methods to determine a reference state are needed. Several methods to compute a reference state of structures without baseline modal information have been developed. Stubbs and Kim [2] 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. [3] applied sensitivity techniques to fit the analytical model and structure using experimental measurements. The Stiffness-Mass Ratios Method [4] 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. In the field of system identification, the Independent Component Analysis (ICA) [5], is a linear method in which the output representation minimizes the statistical dependence of the components of the original representation and captures the essential structure of the data. This method is used herein to develop a representation of the acceleration output acquired by a seismic instrumentation. Fourier transform of this new depiction 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 obtained by conventional spectral analysis with Transfer Functions and Phase Angles of the original acceleration output. In this paper, the Baseline Stiffness Method (BSM) is used 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 to detect loss of stiffness on each element of the structure. 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, a study case from the literature is studied considering the effect of limited modal information and noise. Results are discussed which demonstrate the feasibility of these 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 [5]. 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 T

the random vector consists on finding a linear transform s  Wx such 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 consists of estimating the 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 [6]. 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 [7] code is given in Parra [8], 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 [9] and Cardoso & Souloumiac [10] 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 [11], [12]. This algorithm is parallel and distributed, but is not adaptive. Hyvärinen [13] also showed that when FastICA is used with symmetric decorrelation, it is essentially equivalent to a Newton method for maximum likelihood estimation. 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 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  and their corresponding mode shapes   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 [4] with dimensions 2 s  1 1 :

  u   k1   m1 

 k2     m1 

k   k2      i   m2   mi 

 ki 1   k      s   mi   ms 

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 because the flexural effect is included later on. In this sense, k1 can be determined as:

k1  

12 EI1

(2)

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  2 u4 u5

pi 1 

for j  2,3,...,( i  2 )

(3)

 ki 

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 to

  1 . Thus, when the adjustment by k

the greatest eigenvalue of M Ma

 

1

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. [14], Ka  can be condensed to obtain K a  using the transformation matrix

T  as:

Ka  T  KaT  T

(5)

where  I   Ka11   Ka12    ;  Ka     1 Ka Ka        Ka21   Ka22  22 21   

T   

(6)

For a shear beam building, K  and equation (5) just differ on material properties, specifically, on the magnitude of

 

 

the elastic 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, [15]). 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 [16],

 

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 Y   qqT M  ; Z   qqT K ; q  T M  - 2 ; [ ] is the modal matrix of the structure where H   I   Y  ; 2 and Ω  is a diagonal matrix containing the eigenvalues of the system.

NOISE EFFECT







According to Sohn and Law [17], the mode shape matrix perturbed by noise ˆ is computed as equation 10. This is a common practice to simulate noise effects.

N  ˆ  1  100 R 

(10)



where ˆ is the mode shape matrix without noise, N is a specified noise level in percentage, and R is a normally distributed random number with zero mean and unit variance.

NUMERICAL EXAMPLE。 THREE-STORY FRAME Figure 1 shows the frame building model proposed by Biggs [18]. 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

6

3 8

5

2

7

4.6 m

Floor 1

4

1

Figure 1. Three-story frame (Biggs, 1964)

Damage was simulated reducing the stiffness of element 1 and 8 by 30% and 20% respectively. Figure 2 shows the Fourier spectra of the acceleration output at each floor. It could be observed that in all the floors, two peaks are easily determined, one around 0.79 seconds and the other one around 0.63 seconds. In this case, the period of the input base acceleration is that of 0.63 seconds, thus, it is not associated to the structure. From this figure, only the 0.79 seconds period could be picked, yet, no more information could be drawn. This presents a big problem since more periods are needed to fully represent the dynamical behavior.

Amplitude 100 50 0 80 40 0 60 30 0

Period (sec.) 0.2

0.4

0.6

0.8

1.0

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

In order to solve this problem, ICA was applied to this structure. Figure 3 shows the Fourier spectra for the unmixed sources, extracted with ICA. The signals do not represent the acceleration spectrum at each floor, instead, they represent the spectrum of independent sources of all the acceleration signals obtained from a seismic instrumentation at each floor. Note that it is easy to hand-pick the largest peaks of each independent source. The first source shows two peaks, one at 0.79 and the other at 0.63 seconds. As previously stated, the 0.63 seconds is associated to the excitation, thus is not part of the structural dynamics. The following sources each show a large peak at 0.27 and 0.18 seconds, which altogether represent the structural periods of the frame. It could be observed that ICA gives more information and it is very accesible to extract.

Amplitude 200 100 0 6 4 2 0 0.6 0.4 0.2 0

0.2

0.4

0.6

0.8

1.0

Period (sec.)

Figure 3. Fourier spectra of the unmixed sources, extracted with ICA.

Conventional spectral analysis with Transfer Functions and Phase Angles [19] is later used to extract mode shapes for each of the selected periods. Table 1 shows the periods obtained with the finite element model and the ICA. Also in Table 1, mode shapes are presented which characterize the dynamical behavior of the system. It is important to notice that the third mode shape shows an odd figure, because the expected classical shape would have the middle mass move contrary to the masses above and below, nonetheless, the shape obtained with conventional spectral analysis shows all the masses moving in the same direction. In this case, since heuristics is also part of dynamical analysis, the third mode shape is rearranged, for damage detection purposes.

Table 1. Modal parameters for the three-story frame Mode 1 2 3 Simulated 0.793 0.265 0.179 T (sec.) ICA 0.792 0.265 0.179 1.000 -1.000 -1.000 Mode shape. ICA 0.906 -0.183 -0.799 0.642 0.908 -0.359 Error (%) 0.1 0.0 0.0

The effect of limited modal information was studied applying the BSM to the structure to identify location and severity of damage varying the number of modes from 3 to 1 to determine the baseline state of the plane frame. Results are presented on Figure 4. It can be observed that location of damage was identified for element 1 even for the case when just one mode was used. Damaged element 8 was identified when three or two modes were utilized. The proposed method herein identified the damaged elements when the first two modes were used.

Loss of stiffness (%) 100 80 60 40 20 0

1

2

3

4

5

7

6

8

9

Number of element

Simulated Three modes Two modes One mode Figure 4. Degradation of stiffness using the BSM applied to the three-story frame.

Regarding to loss of stiffness, Table 2 presents error values between simulated and computed damage magnitudes from Figure 4. Note that for the case of complete modal information the proposed method determined damage magnitude with error values less than 10%, acceptable in engineering. When two modes are considered, the error value for element 1 was smaller than 10%; however, for element 8, damage magnitude was not adequately determined; neither for the case when just one mode was used.

Table 2. Error values, in percentage, between simulated and computed damage magnitude Number of modes used Damaged Element 3 2 1 1 -3.3% -4.3% 221.5% 8 4.1% -79.2% ---

In order to include the effect of noise in the damage detection process, the extracted modal parameters from the damaged structure were perturbed according to equation 10 for 2, 4, 6 and 50% level of noise. Figure 5 shows the error values between computed damage magnitudes with and without noise when three modes were used to determine the baseline state of the structure. It can be noted that error values are smaller than 10% for all simulated noise levels except for 50%. Error value (%) 10 5 0

Number of element

1 2

3

4

5

6

7

8

9

-5 -10 -15 0% 2% 4% 6% 50% Figure 5. Error values between computed damage magnitude with and without noise, utilizing three modes.

Figure 6 shows the error percent values between computed damage magnitude with and without noise for element 8 when two modes were used to determine the baseline state of the structure. It can be observed that the

greater the level of noise, the greater the error. Based on this figure and using interpolation, error values are smaller than 10% when noise levels in the computed mode shapes are smaller than 24%. Error value (%) 14 12 10 8 6 4 2 0 0

10

20

30

40

Level of noise (%) 50

Figure 6. Error values for Element 8, computed for damage detection with and without noise, utilizing two modes.

Error values for element 1 were 0% which is indication that the BSM is less sensitive to noise effects for columns than for beams. This is an advantage related to the strong column-weak beam concept, when applicable. This was corroborated with the same result when just one mode was utilized.

CONCLUSIONS The use of ICA to select the natural vibration periods of a structure has proven a powerful tool due to the fact that the Fourier spectra of the unmixed signals shows clearer peaks for the selection process. As for the BSM if a sufficient number of mode shapes are used, location and severity of damage, in terms of loss of stiffness from a structure without baseline references, could be determined. When only a limited number of modes were used to fit the damaged stiffness from dynamic measurements, the BSM was capable to determine the location of damage. When noise is introduced to the computed mode shapes, an acceptable error value is achieved when a level of noise is less than 20% in this case. Finally, it could be noted that the BSM is sensitive to noise in mode shapes when damage is to be detected in beams, thus, hinting to the fact that damage in beams should be detected with the lesser amount of noise in mode shapes as possible.

ACKNOWLEDGEMENT The authors would like to gratefully acknowledge the Instituto Politécnico Nacional for supporting this research project, No. 20080145.

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