Damage identification for structural health monitoring using fuzzy pattern recognition

Engineering Structures 27 (2005) 1774–1783 www.elsevier.com/locate/engstruct

Damage identification for structural health monitoring using fuzzy pattern recognition M.M. Reda Tahaa,∗, J. Lucerob a University of New Mexico, Albuquerque, NM 87131, USA b Los Alamos National Lab, MS T001, Los Alamos, NM 87545, USA

Available online 25 July 2005

Abstract Uncertainty abounds with in situ structural performance assessment and damage detection in Structural Health Monitoring (SHM). Most research in SHM focuses on statistical analysis, data acquisition, feature extraction and data reduction. We introduce a method to improve pattern recognition and damage detection by supplementing Intelligent Structural Health Monitoring (ISHM) with fuzzy sets. Intuitively we know that damage does not occur as a Boolean relation (one of two values, true or false) but progressively. Bayesian updating is used to demarcate levels of damage into fuzzy sets accommodating the uncertainty associated with the ambiguous damage states. The new techniques are examined to provide damage identification using data simulated from finite element analysis of a prestressed concrete bridge without a priori known levels of damage. © 2005 Elsevier Ltd. All rights reserved. Keywords: Structural health monitoring; Artificial neural network; Wavelet multi-resolution analysis; Damage index; Fuzzy set; Bayesian updating

1. Introduction The emerging field of structural health monitoring (SHM) addresses the in situ behavior of structures by assessing their performance and recognizing damage or deterioration. SHM involves system state definition, data acquisition, data filtration, feature extraction, data reduction, pattern recognition and decision making. Each of these components is equally important to determine the state of health of a structure. However, the bulk of research on SHM has been developed over the last decade on data acquisition, feature extraction and data reduction techniques. We propose to incorporate new theories of uncertainty with recent developments in artificial intelligence and digital signal processing for a more robust SHM system. The Wavelet Transform (WT) has many desirable features that become the motivation for its use over the more restrictive Fourier Transform (FT) in engineering systems [1,2]. These features have proven themselves to be useful ∗ Corresponding author. Tel.: +1 505 277 1258; fax: +1 505 277 1988.

E-mail address: [email protected] (M.M. Reda Taha). 0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2005.04.018

for information discernment in digital signal processing (e.g. inertial navigation [3] and biomedical engineering [4]). As such, the WT lends itself naturally to damage detection and recognition. For example, Paget et al. [5] exploits the WT’s qualities for damage assessment for aerospace composites. Similarly, the WT is used for crack detection in sewer pipes [6], fault detection in DC electro motors [7], and damage detection in prestressed concrete specimens [8]. Hence, the use of the WT in damage recognition algorithms has been significantly increasing [9,10]. Reda Taha et al. [11] proposed integrating artificial neural networks (ANN) and wavelet multi-resolution analysis (WMRA) for intelligent structural health monitoring (ISHM). We build on ISHM and address two questions that are encountered in SHM analysis: “Is there damage in the structure?” and, if yes, “How severe is this damage?” These two questions encompass a variety of uncertainties, for instance, rates and types of loading: slowly, (environmental and time-dependent conditions), quickly and predictably, (heavy traffic loading), and quickly and unpredictably (earthquake acceleration). This vagueness in loading propagates to damage classification as non-distinct

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

and overlapping levels of severity such as minor, moderate, severe and very severe. Statistical pattern recognition can account for only distinct damage features for which data can have a crisp point-wise quantification. Thus using statistical pattern recognition, unfortunately, we immediately restrict the capabilities of the SHM system to the limitations of probabilistic assumptions. Biondini et al. [12] presented a good discussion on uncertainties encountered in structural response and the distinctions between various types of information as random, fuzzy, etc. A major challenge in damage recognition in SHM is that the sample space in which the damage feature is defined is non-stationary; it is dependent on the health state of what is assumed to be a healthy structure. It was also observed that the frequency of occurrence of a healthy state in the database is not constant from one set of observations to another. Therefore, it becomes obvious that damage cannot be classified as a random process, and consequently probabilistic assumptions for damage recognition cannot hold. As such, conceiving a strict probabilistic mathematical abstract of the system might lead to misleading uncertainty analysis. We assert that uncertainty in damage recognition is more epistemic (subjective) than aleatoric (objective); that is to say that we can improve our knowledge about damage and reduce the uncertainty of its recognition as we obtain further observations that are not limited to measurements alone. The major difference between aleatoric and epistemic uncertainties is that aleatoric refers to random uncertainties that are irreducible while epistemic uncertainty is related to system knowledge and thus is a reducible uncertainty. This paper extends the ISHM system to associate damage with particular levels of damage. Furthermore, we are working towards a resolution of the emerging challenge of coupling both aleatory (random) and epistemic (system knowledge) uncertainties to develop fuzzy sets for improved pattern recognition and decision making. Specifically, we address the challenge of using both dense and sparse data. We propose training a neural network on the vibration responses of a structure under continuous arbitrary loading for a specified time period. This time period is used to obtain as much information as possible on the typical loading response of a healthy structure. From this data set, we will infer damage states. Future events or knowledge will then be used to update the assessments yielding a reliable recognition of damage. This represents a novel approach by progressively accounting for uncertain information given a limited set of statistics. These statistics are measured single point values. We can make good use of the aggregate statistics as dual information, single point and interval elements. The single point elements will be used to establish the pattern of healthy performance. Furthermore, the interval elements as evidence for more serious damage will establish their associated linguistic damage levels, “Little”, “Moderate”, and “Significant”. These levels will be defined as fuzzy sets. As such, they will be used to recognize

1775

the most similar observation state as healthy or damaged. Establishing such fuzzy damage states will provide a reliable means of damage pattern recognition that not only can be used for damage warning but can also be related to structural reliability, safety and surety [13].

2. Three stages of SHM Our method parallels the intelligent paradigm proposed by Worden and Delieu-Barton [14]. Their method for an intelligent fault detection system is a hierarchical analysis consisting of progressive levels of health: defect, damage and fault. The three stages that they describe to assess a system are anomaly detection, feature extraction, and pattern processing. However, in this approach, both unsupervised and supervised trainings are required for anomaly detection and damage identification respectively [14]. We propose a different approach that combines Neural and Fuzzy unsupervised training algorithms to detect damage occurrence and to classify progressive levels of damage. We use a neuralwavelet approach for damage occurrence detection after a sufficient unsupervised training period. Next, we partition the potential damage levels into fuzzy sets using Bayesian updating. Finally, we recognize a particular level of damage associated with a new input observation vector using a technique in fuzzy systems theory, the similarity metric. The following sections describe in detail the theory and implementation of each of the three stages. 2.1. Damage detection using the wavelet norm index The proposed SHM system combines the WT with ANN to provide intelligent digital signal processing [11]. The WT for a discrete time signal x(n) can be described as
x(n)ψ(2− j n − k) (1) C j,k = 2(− j/2) n

where ψ(n) is the wavelet function utilized in the wavelet transform and 2(− j/2)ψ(2− j n − k) are scaled and shifted versions of the wavelet function based on the values assigned for scaling and shifting ( j is the scaling coefficient and k is the shifting coefficient). The j and k coefficients take integer values for different scaling and shifted versions of ψ(n). C j,k represents the corresponding wavelet coefficients that are similar to the FT coefficients [1]. Wavelet multi-resolution analysis (WMRA) is a technique used to perform the discrete wavelet transform. It allows the decomposition of signals into various resolution levels. The data with coarse resolution which contains information about low frequency components and retains the main features of the original signal is detached from the original signal and is named as “the approximations”. The data with fine resolution which contains information about the high frequency components is detached and is named as “the details” [15,16].

1776

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

Consider j and k to be the scaling and shifting indices respectively. Each value of j corresponds to analyzing a different resolution level of the signal. The approximation coefficient a j,k at the j th resolution of an input digital signal x(n) can be described as
a j,k = 2(− j/2) x(n)φ(2− j n − k) (2) n

where φ is the scaling function. Scaling functions are similar to wavelet functions except that they have only positive values [15]. The approximation signal f j (n) at the j th resolution level can be computed as ∞


f j (n) =

a j,k φ j,k (n).

(3)

k=−∞

The details coefficient d j,k at the j th resolution level and the detail signal g j (n) are then computed as
x(n)ψ j,k (n) (4) d j,k = n

g j (n) =

∞


d j,k ψ j,k (n)

(5)

the combined WMRA and ANN form the damage detection module. This module makes it possible to extract features from signals in the time domain, to pattern those features and then recognize the differences for damage identification. A schematic representation of the proposed damage detection module is presented in Fig. 1. A new observed signal (Sensor-Measurement) is decomposed using WMRA into a number of measured signal approximations (SAMi ) and a number of measured signal details (SDMi ) with 1 < i < J . In the meantime, the predicted signal (ANNPredicted) is also decomposed using WMRA into a number of predicted signal approximations (SAPj ) and predicted signal details (SDpi ) with 1 < j < J with J defined as the level of decomposition where the wavelet analysis is to be terminated. An error signal representing the error between the predicted and the measured signal is computed in the wavelet domain by subtracting the wavelet coefficients of the predicted signal from the corresponding wavelet coefficients of the measured signal as in Eq. (8): Cec = Cac − Csc

(8)

k=−∞

where ψ j,k (n) is the wavelet basis function. If the process is repeated for the j + 1 resolution level, thus decomposing the approximation signal f j (n), the original signal x(n) can be reconstructed using an infinite number of details signals at infinite resolution levels as ∞ ∞ ∞


x(n) = g j (n) = d j,k ψ j,k (n). (6) j =−∞

j =−∞ k=−∞

If the analysis stopped at the Pth resolution level, the original signal can be approximately reconstructed using the approximation at the Pth level and all the details starting from the first level up to the Pth level as x(n) =

∞
k=−∞

a P,k · φ P,k (n) +

P
∞


d P,k · ψ P,k (n). (7)

j =1 k=−∞

The first term represents the approximation at level P and the second term represents the details at level P and lower [10,16]. During a time period of healthy structural performance (THealthy) the neural network is trained to map the relationship “features” of the healthy structural dynamics after being decomposed using WMRA. ANN are used to learn both of the approximation and detail parts of the decomposed signals from interspersed locations along the structure. The Leveneberg–Marquardt training algorithm is used to train the network (i.e. achieving minimum mean square estimation error) [17,18]. A significantly low training error (<1E-5) is used to guarantee the ability of ANN to capture the local relationship between the structural dynamic responses of these remote parts. The training process is performed during a time of healthy performance and therefore the ANN are built to simulate healthy structural responses. At a time of unknown structural performance,

where Cec , Cac and Csc represent vectors of wavelet coefficients for error, measured and predicted signals respectively. It has been argued that an energy index that is related to damage in the structure can be established by computing the energy of the error signal [11]. As Parseval’s theorem relates the norm of the wavelet coefficients to the energy of the signal [2], a damage metric called the wavelet norm index (WNI) is produced as a measure of the energy of the error signal. The WNI is a dimensionless index as it is computed in the wavelet domain. The WNI at any instance N (WNI N ) can be represented in terms of the wavelet coefficients of the error signal (or measured and predicted signals) as WNI N = Cec 2

(9)

where Cec  is the norm of the wavelet coefficients of the error signal. Thus WNI N can be computed as   J


2 2 (10) WNI N = |a( J )ec(k)| + |d( j )ec(k)| k

k

j =1

where k is the number of samples observed during a specific time period. The difference between the damaged and healthy conditions can thus be established by evaluating the WNI that represents the energy of the error signal at various operating instances. It can also be argued that the larger the difference the higher the level of damage in the structure. As the neural-wavelet module is trained to simulate healthy system dynamics, the WNI of the error signal shall be relatively constant during healthy operations. The presence of new superimposed dynamic components will account for an increase in the difference between the predicted and measured signals indicated by the WNI of the error signal. This change in the WNI would suggest an occurrence of damage. It has been shown that changes

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

1777

Fig. 1. A schematic representation of the damage detection module for SHM that augments ANN and WMRA.

in the WNI can be linked to damage by observing the probability of the WNI derived from subsequent healthy instances [11]. 2.2. Establishing fuzzy health patterns We have just presented the procedure for damage quantification. Given the inherent uncertainty in the damage index and admitting a level of imprecision in the damage states using fuzzy sets, we can feasibly define damage levels. We propose a new method to determine damage levels by establishing fuzzy sets on the wavelet norm index (WNI). We combine sparse measured point data and interval data, Bayesian updating, and expert judgment within the construct of the fuzzy logic framework to substantiate fuzzy sets for damage assessment. This allows us to combine both aleatory and epistemic uncertainty. After a training period, a vector of the WNI representing consecutive measurements of vibration responses will be compared to defined fuzzy damage levels. The state of the structure will then be determined based on the vector’s (fuzzy pattern) degree of similarity to the defined fuzzy damage levels. This current approach expands on the previous all-probabilistic method in [11] where multiples of standard deviations away from the “healthy” training information were used to define progressively severe damage levels. The disadvantage of the all-probabilistic approach is that it is being constrained with many assumptions. One assumption is the assignment of a

symmetric distribution to the response levels. Without an infinitely number of measurements, this cannot be proven to be the case. Where the luxury of having an infinitely number of measurements or performing extensive computer simulation is not feasible, pragmatic damage recognition becomes the answer. What we can do is use the existing information as prudently as possible. This means that we will use measured data along with expert judgment to tailor the fuzzy sets using both dense and sparse data. We will have ample data during the healthy training period. Hence we will develop a “Healthy” fuzzy set that is Gaussian shaped and reflects an undamaged structural state. Consequently, for infrequent or sparse damage data we will develop fuzzy sets that mimic the shape of a probability distribution suited for sparse data, a Poisson distribution function. The Poisson distribution is useful for cases where infrequent events are seen; it essentially describes a counting process. We are taking a conceptual departure from this and using only the shape of the Poisson distribution over a continuous domain of expected values as in [19]. Huyse and Thacker [19] presented how to overcome the challenge of conflicting and insufficient data which is prevalent in a wide variety of applications such as risk and damage assessment in engineering. In our application, infrequent and high WNI values might be measured and the shape of the distribution (membership function) can be continually updated. This summarizes the data frequency and gives us a

1778

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

well justified value on which to define our damage sets. This proposed approach is much more convenient in that it solely depends on inference of healthy observations to provide self-referential damage recognition without demanding any further knowledge via structural simulation or special field testing. Here, four structural health patterns (damage levels) ranging from healthy to significantly damaged are proposed. The non-distinct boundaries between these health patterns and the inherent overlap therewith make fuzzy systems a suitable candidate for damage pattern recognition. We begin by defining the “Healthy” pattern. Observations are made during a time period, THealthy, of typical performance. Here, a reasonable amount of measurements exists to determine the healthy fuzzy set. Therefore the structural health membership function is defined as a left-shouldered fuzzy set using the Gaussian function described in Eq. (11) in a similar fashion to damage functions used in damage analysis for earthquakes [20,21]:    exp −(x − WNI H ) x ≥ WNI H 2 (11) µ H (x) = σWNI H   ˜ 1 x < WNI H where µ H (x) represents the membership function of the fuzzy set˜ representing a healthy pattern that has a mean observed wavelet energy index of WNI H and a standard deviation of σWNI H . Furthermore, information from this first fuzzy set can be used to develop the proximate fuzzy set, “Little Damage”. We begin with a tentative cognizance of the domain “Little Damage”. A desirable feature of multiple fuzzy sets is complete coverage over the universe of discourse. Our universe of discourse is the set of all possible WNI values, non-negative and real numbers. Here we expand the universe of discourse for the possible fuzzy sets beyond the observations merely based on the presumption that all training has been performed during healthy performance period. Since the domain of this first set, “Healthy”, is known, the lower bound of “Little Damage” can be located. Specifically, the value, WNI H , can be assumed equal to the lower bound of the “Little Damage” fuzzy set such that Little Damage = {x | x ≥ WNI H }. In the parlance of probability, we assign to “Little Damage” fuzzy set a non-informative prior distribution to describe our initial knowledge and shape of the fuzzy set. A non-informative prior distribution attempts to represent a certain level of initial ignorance about the system [22]. Unless specific knowledge about the system is available, a uniform distribution is traditionally assigned to describe the non-informative prior. In our case of damage recognition using the WNI values, we realize that the fuzzy sets are constructed in such a way that each fuzzy set will cover its own range of WNI values while reserving an overlap with the other fuzzy sets. This knowledge negates the possibility of using uniform distribution to represent the non-informative prior. Therefore, Jeffrey’s non-informative

prior distribution [22] over the domain X (WNI values) that assumes no-observation, x, is used here as described by Eq. (12): Jeffrey’s non-informative prior density (no observation): 1 (12) p(X) = √ . X Therefore, for the three damage levels beyond the “Healthy” set, a similar procedure is used to develop their initial estimates in accordance with our knowledge of fuzzy set boundaries. Subsequently, as information (observations) about the structural response becomes available, these Jeffrey’s priors will update the membership function by mimicking the shape of the Poisson density function, Eq. (13). Xx exp(−X). (13) x! Herein lies the essence of our conceptual departure. We are calculating the degree to which data x is contained in the domain X, i.e. x ∈ X. Our approach goes a step further in accommodating uncertainty by using interval data, [x 1 to x 2 ]. Thus, the degree to which the interval is contained in the domain X is calculated using Bayesian updating as explained below. The use of the Poisson density function to express the likelihood of damage is attributed to the fact that the impetus for damage events represents non-frequent occurrences. Poisson distribution is usually used to represent non-frequent and independent random variables [22]. Bayesian Updating is a natural consequence of Bayes’ Theorem. Bayes’ Theorem simply combines prior knowledge about a parameter with additional support data to compute the subsequent knowledge of the parameter. This updated knowledge is known as the posterior distribution and is proportional to the product of the likelihood and the prior distribution as Poisson density function : f (x | X) =

(X | x) · f (X) . f (x | X) =

(X | x) · f (X)

(14)

Eq. (14) shows that the likelihood function, (X | x), which represents the support of X given data point x, combines with the probability function, f (X), to yield new composite knowledge f (x | X). In our application, the likelihood (or support) of X for the interval data [x 1 , x 2 ], is denoted (X | x) and can be represented by Eq. (15),  x2 f (x | X)dx. (15) (X | [x 1 , x 2 ]) = x1

Substituting Eq. (15) into Bayes’ Theorem, Eq. (14), for interval data gives

x f (X) x12 f (x | X)dx

x f ([x 1 , x 2 ] | X) =

. (16) f (X) x12 f (x | X)dx dX Moreover, substituting Eq. (13) for the Poisson distribution into Eq. (15) yields the likelihood given interval data [x 1 , x 2 ] as

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

Interval likelihood function (strength of fuzzy set): x2
Xx exp(−X). f ([x 1 , x 2 ] | X) = x=x 1 x!

1779

For subsequent interval observations, the posterior density function becomes

requires a concept in fuzzy logic, degree of similarity. The degree of similarity of an input observation vector measured at any time period N will be measured and compared to the established fuzzy sets in order to determine the structure’s health. Ross [21] introduced a metric parameter to represent the degree of similarity “D” between two fuzzy vectors A ˜ and B as ˜ 1 (21) D = [(A • B ) + (A ⊕ B )] 2 ˜ ˜ ˜ ˜ where (A • B ) represents the inner product of the two fuzzy ˜ B and (A ⊕ B ) represents the outer product vectors ˜A and ˜ ˜ [21].˜ The˜ similarity metric “D” can be of the two vectors computed to represent the degree of similarity between the new vector of WNI observations representing the unknown structural health and each of the four fuzzy structural health patterns previously defined. Using principles of the maximum approaching degree explained in [21], the fuzzy structural health pattern with the maximum similarity metric “D” will be the closest health pattern to describe the structural health. A case study describing this procedure is explained below.

Posterior density function (multi observations): x2 Xx exp(−X) x! ∗ f i−1

3. Case study: Damage identification in a simulated prestressed concrete bridge

(17)

The interval likelihood function, which can be thought of as the strength of the fuzzy set, Eq. (17), can be used to update the membership function f (x | X) in a Bayesian sense producing the posterior density function. This is achieved by substituting Eq. (17) in Eq. (16) while including Jeffery’s non-informative prior (Eq. (12)) to represent the membership function f (x | X) in the first update for the case of one single observation (Eq. (18)). The non-informative prior is therefore used only once. Posterior density function (single observation): x2 X x−1/2 exp(−X) x! f 1 ([x 1, x 2 ]1 | X) =

x=x 1

 x 2 Γ x+ 1 2

(18)

x!

x=x 1

x=x 1

f i ([x 1 , x 2 ]i | X) =

.

x2 Γ (x) x=x 1

(19)

x!

where [x 1 , x 2 ]i is the i th interval observation. Additionally, the last fuzzy set, “Significant Damage”, will be assumed to act as the upper bound on the universe of discourse much like “Healthy” such that it will be rightshouldered as  Eqs. (18) and (19) x < WNI∗S (20) µ S (x) = 1 x ≥ WNI∗S ˜ where WNI∗S is the prototypical “Significant” WNI value. To summarize, WNI values over a time period will be used to develop the fuzzy set, “Healthy”. In succession, the lower bounds for the remaining fuzzy sets will use the shape of Jeffrey’s non-informative prior in posterior updating of the likelihood represented by Poisson distribution. Interval data within the bounds set forth by experts will update the functions in Eqs. (18)–(20) to form three fuzzy damage sets, “Little Damage” “Moderate Damage” and “Significant Damage”. This is an ampliative process that incorporates both single point and interval data to build on received evidence in the form of statistical data or expert judgment. These newly developed fuzzy sets are now ready to be used to recognize recent observations. 2.3. Fuzzy pattern recognition Now we want to recognize (identify, classify) a set of consecutive input observations (WNI) into one of the predefined fuzzy set levels of damage. This recognition

To demonstrate the ability of the proposed wavelet-aided fuzzy damage identification to perform structural health monitoring, a 3D finite element (FE) model of a 40 m span simply supported, prestressed concrete bridge was developed. The bridge model consisted of two prestressed concrete girders spaced at 4.0 m and a reinforced concrete deck slab. Each girder was modeled with 20 beam elements, each of 2.0 m in length requiring 21 nodes each with four degrees of freedom (DOF). The deck slab was modeled as discrete beam elements connecting the two girders. The girder has a trapezoidal cross section with a 1.65 m depth and a mass per unit length of 4800 kg/m. The deck slab has a rectangular cross section with 200 mm thickness. The bridge concrete has a characteristic compressive strength of 45 MPa with a modulus of elasticity of 34 700 × 103 MPa. The analysis assumed linear elastic behaviour of the concrete. A schematic of the bridge and the finite element model are presented in Figs. 2 and 3. Thirty accelerometers were assumed to be mounted on the two girders to monitor accelerations due to traffic loading. The sensors are denoted S1, S2 to S30 as indicated on Fig. 3. The number of sensors significantly affected the accuracy of the model. Although model sensitivity to the sensors’ number and location was not part of this study, it is evident that large sensor arrays are capable of providing sufficient details about regional behaviour under loading and thus make it possible to identify patterns of dynamic behaviour from different regions of the bridge. The output signals of these accelerometers are obtained from the finite element model as the z-axis acceleration component at the nodes located at the

1780

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

Fig. 2. A schematic representation of the 40 m span prestressed concrete bridge used in the case study.

represents the fuzzy set “Little Damage” using Jeffery’s noninformative prior along with no observation. The simulation process is then developed and the membership functions were updated using Eq. (20). The final shape of all four fuzzy sets, “Healthy” “Little Damage” “Moderate Damage” and “Significant Damage”, are shown in Fig. 6(a).

Table 1 WNI observations vector measured during period of healthy performance THealthy (H denotes a known healthy instance) Instance

H1

H2

H3

H4

H5

H6

H7

H8

H9

WNI

53.53

78.8

40.92

40.1

56.24

35.5

40.1

63.5

53.1

Fig. 3. A schematic representation of the FE model for a prestressed concrete bridge.

4. Results and discussion accelerometer positions. Regional input and desired signals are simulated for training during healthy performance and for testing during unknown health instances using the finite element model. The analysis assumed linear elastic behavior of concrete as the target was to identify damage represented by cracking and loss of stiffness. The bridge response was recorded during healthy operational time due to different truck loads passing over the bridge with different speeds ranging from 30 to 80 km/h. Sample accelerations representing the typical bridge response from three distant regions are presented in Fig. 4. These responses were used to train a neural network designed to predict the system dynamics as previously explained. A healthy time period, THealthy, is assumed and typical performance was represented by nine consecutive WNI values which comprised the observation set. Table 1 presents the WNI observations vector representing “Healthy” performance. We define X = [0–150] in Eqs. (18)–(20) to encompass all the fuzzy sets domain of WNI values. This domain includes the specific interval observation data which will be used to develop the individual WNI domains, X Healthy, X Little, X Moderate , and ˜ X Significant as the four fuzzy ˜structural ˜health patterns. Fig. 5 ˜ shows the first estimate of the membership function that

The system was tested to predict the damage performance of the bridge due to a 430 kN truck passing with a speed of 54 km/h. The damage resulted by incorporating cracking at mid-span of the concrete girder modeled as a significant reduction of stiffness (40% stiffness reduction). All the simulated signals were augmented first by adding a random noise signal. The signals were then directed to the damage diagnostic module as a set of signals representing unknown health condition. The signals were therefore first decomposed using WMRA. The error signals were calculated at each instance and the WNI vector of observations was computed. Table 2 presents the WNI vector of the unknown health condition. Fig. 6(b) shows the four fuzzy structural health patterns along with the observation fuzzy set. Eq. (21) is then used to recognize an observation set as one of these four health patterns. The similarity metrics (D1, D2, D3, D4) of the testing instance with respect to the four structural health fuzzy patterns (Healthy, Little Damage, Medium Damage and Significant Damage) were computed. A summary of values of the similarity metric “D” for this testing instance is presented in Table 3. Using the principles of the maximum approaching degree the damage testing instance for this case approaches the maximum similarity at D4 = 1.0, indicating

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

1781

Fig. 4. Sample of accelerations from three sensors (S1, S2 and S19) representing the bridge response to traffic as simulated using the finite element model.

significant damage occurrence in the prestressed concrete bridge. Bayesian updating was introduced to an already novel method for structural health monitoring, ISHM, for the purpose of defining fuzzy sets that represent particular levels of damage. Consequently, these fuzzy sets were used in a similarity metric to assess real-time response measurements. This fuzzy pattern recognition allowed for the characterization of structural health without forcing a priori knowledge assumption. Instead we use sparse data to support expert knowledge to detect damage. Table 2 WNI observations vector representing unknown health performance (U denotes an unknown healthy instance) Instance

U1

U2

U3

U4

U5

WNI

118.4

81.6

92.7

138.8

67.3

Table 3 Similarity metric “D” computed for the testing instance with respect to the four fuzzy structural health patterns Health pattern

Healthy

Little Damage

Moderate Damage

Significant Damage

Similarity metric

D1 0.511

D2 0.506

D3 0.664

D4 1.00

To summarize, Bayesian updating was introduced to an already novel method for structural health monitoring, ISHM, for the purpose of defining fuzzy sets that represent particular levels of damage. Consequently, these fuzzy sets were used in a similarity metric to assess real-time response measurements. This fuzzy pattern recognition allowed for the characterization of structural health without forcing a

Fig. 5. “Healthy” fuzzy set and “Little Damage” fuzzy set as estimated using Jeffery’s non-informative prior with no observations and prior to performing Bayesian update.

priori knowledge assumption. Instead we use sparse data to support expert knowledge to detect damage. In addition, we included a concurrent effect as random noise that could adulterate the response damage detection. This noise is representative of environmental conditions like wind and temperature effects. In fact, Ko et al. [23] pointed out that these concurrent effects could have a substantial influence on modal variability potentially leading to a false positive damage assessment. Our approach would side step this complication in two respects. First, we are not using a modal analysis approach. Our approach simply relies on acceleration prediction and measurement comparison. The environmental conditions would be embedded in the healthy response over the training period. And since we are projecting a non-stationary damage assessment, slower environmental effects like shrinkage and cracking concrete can update the damage fuzzy sets. Secondly, using WMRA

1782

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783

feature that can be used; any potential damage feature can be used. Consequently, this method and metric are generic for any structural health monitoring system. Thus it becomes an approach with far-reaching potential as it is independent of the feature used to represent damage. 5. Conclusions We have demonstrated a method to quantify evidence of damage levels in structures by means of the computations of fuzzy set theory. Accelerations from sensors distributed over the bridge are analyzed using a wavelet-neural network module to establish patterns of dynamic behaviour of the bridge. A damage metric computed in the wavelet domain is established. The damage metric describes the energy of the error signal representing the error between the measured signal and the signal predicted by the neural network representing the healthy pattern performance. The proposed method uses Jeffery’s non-informative prior in a Bayesian updating scheme to infer fuzzy health “or damage” patterns. A case study to identify damage occurrence in a prestressed concrete bridge has been discussed. The model has been shown to be capable of identifying damage accurately. Acknowledgments

Fig. 6. The final shape for the fuzzy structural health patterns “Healthy”, “Little Damage”, “Moderate Damage” and “Significant Damage” after Bayesian update. (a) The four fuzzy sets. (b) The four fuzzy sets along with unknown health performance.

allows a level by level analysis unlike the FT in Ko et al. [23]. We can find the optimum level or combination thereof to discern service load damage from its wind load contribution. For example, during a certain time of the year when the temperature affects the structure considerably, it might be more beneficial to use a particular set of levels. Hence the proposed model has a variety of perspectives to include in separating erroneous environmental influences, yet keeping track of slow and progressive environmental caused damage. Unlike similar approaches provided by [14] and [24], our method does not need supervised training or retraining of any sort. All the damage levels are defined with an uncertainty tolerance during the initial unsupervised training period. Using only healthy observation data, we constructed fuzzy health (or damage) patterns for a case study of a simulated prestressed concrete bridge. Subsequently, a similarity metric was used to identify a new set of observations into a particular level of damage based on the WNI as a damage index. However, the WNI is not the only

This research is funded by Sandia National Laboratories (SNL). Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC0494AL85000. The authors gratefully acknowledge this support. The financial support to the second author by Los Alamos National Laboratory is greatly appreciated. References [1] Hubbard BB. The world according to wavelets, The story of a mathematical technique in the making. 2nd ed. Natick (MA): A. K. Peters; 1998. [2] Strang G, Nguyen T. Wavelet and filter banks. New Yok (NY): Wellesley-Cambridge Press; 1997. [3] Noureldin A, Osman A, El-Sheimy N. A neuro-wavelet method for multi-sensor system integration for vehicular navigation. Journal of Measurement Science and Technology 2004;15:404–12. [4] Nishikawa D, Yu W, Yokoi H, Kakazu Y. Analyzing and discriminating EMG signals using wavelet transform and real-time learning method. Intelligent Engineering Systems through Artificial Neural Networks 1999;9:281–9. [5] Paget C, Grondel S, Levin K, Delebarre C. Damage assessment in composites by Lamb waves and wavelet coefficients. Smart Materials and Structures 2003;12:392–402. [6] Browne M, Dorn M, Ouellette R, Christaller T, Shiry S. Wavelet entropy-based feature extraction for crack detection in sewer pipes. In: Proc. of the 6th international conference on mechatronics technology. 2002, p. 202–6. [7] Boltežar M, Simonovski I, Furlan M. Fault detection in DC electro motors using the continuous wavelet transform. Meccanica 2003; 38(2):251–64.

M.M. Reda Taha, J. Lucero / Engineering Structures 27 (2005) 1774–1783 [8] Melhem H, Kim H. Damage detection in concrete by Fourier and wavelet analysis. Journal of Engineering Mechanics, ASCE 2003; 129(5):571–7. [9] Farrar CR, Jauregui DA. Comparative study of damage identification algorithms applied to bridge. Smart Materials and Structures 1998;7: 704–19. [10] Reda Taha MM, Noureldin A, Lucero J, Baca T. Wavelet transform for structural health monitoring: A compendium of uses and features. Journal of Experimental Mechanics 2005 [in review]. [11] Reda Taha MM, Noureldin A, Osman A, El-Sheimy N. Introduction to the use of wavelet multiresolution analysis for intelligent structural health monitoring. Canadian Journal of Civil Engineering 2004;31: 719–31. [12] Biondini F, Bontempi F, Malerba PG. Fuzzy reliability analysis of concrete structures. Computers and Structures 2004;82:1033–52. [13] Frangopol DM, Neves LC, Petcherdchoo A. Health and safety of civil infrastructures: A unified approach. In: Proceedings of the 2nd workshop on structural health monitoring of innovative civil structures. 2004, p. 253–64. [14] Worden K, Delieu-Barton JM. An overview of intelligent fault detection in systems and structures. Structural Health Monitoring 2004;3:85–98. [15] Chui CK. Wavelets: A tutorial in theory and application. New York (NY, USA): Academic Press; 1992. [16] Mallat S. A wavelet tour of signal processing. 2nd ed. London (UK):

1783

Academic Press; 1999. [17] Jang J-SR, Sun C-T, Mizutani E. Neuro-fuzzy and soft computing, A computational approach to learning and machine intelligence. Englewood Cliffs (NJ, USA): Prentice Hall Inc.; 1997. [18] Haykin S. Neural networks: A comprehensive foundation. Englewood Cliffs (NJ, USA): Prentice Hall Inc.; 1999. [19] Huyse L, Thacker BH. A probabilistic treatment of conflicting expert opinion. In: 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference. 2004. [20] Reda Taha MM, Noureldin A, Ross T. A fuzzy-aided damage recognition for intelligent structural health monitoring. In: Proc. of the second European workshop on structural health monitoring. 2004, p. 669–76. [21] Ross TJ. Fuzzy logic with engineering applications. 2nd ed. Chichester (UK): John Wiley & Sons; 2004. [22] Box G, Tiao G. Bayesian inference in statistical analysis. John Wiley & Sons; 1992. [23] Ko JM, Chak KK, Wang JY, Ni YQ, Chan THT. Formulation of an uncertainty model relating modal parameters and environmental factors by using long-term monitoring data. In: Liu S-C, editor. Smart materials and structures, 2003: Smart systems and nondestructive evaluation for civil infrastructures, SPIE 5057. 2003, p. 298–307. [24] Ni YQ, Wang BS, Ko JM. Constructing input vectors to neural networks for structural damage identification. Smart Materials and Structures 2002;11:825–33.