bridge integrity assessment

International Journal of Structural Stability and DynamicsVol. 9, No. 1 (2009) 11–43c© World Scientific Publishing Company

BRIDGE INTEGRITY ASSESSMENTBY CONTINUOUS WAVELET TRANSFORMS

A. ALVANDI∗, J. BASTIEN, E. GREGOIRE and M. JOLIN

Research Center on Concrete Infrastructure (CRIB)Department of Civil Engineering, Universite Laval

Quebec, G1V0A6, Canada∗[email protected]

Received 9 January 2008Accepted 16 May 2008

The potential of continuous wavelet transforms for damage assessment of existing bridgesis investigated herein. Different types of continuous wavelet transforms have been underinvestigation and the most effective ones have been introduced in a toolbox to auto-mate the damage assessment procedure. In this paper, the performance of the waveletapproach and the influence of different parameters in the damage assessment proce-dures are studied through two examples: a simply supported beam and a three-spanconcrete bridge. Applying the wavelet transforms to a structure’s static and/or dynamicresponse showed promising results with regard to localization of structural modificationor damage. This paper underlines the high sensitivity of the wavelet analysis to damageintensity and its ability to be applied directly to the damaged data. These key char-acteristics could lead to this approach becoming one of the best for structural healthmonitoring of existing bridges in the near future.

Keywords: Wavelet analysis; damage assessment; structural health monitoring; dynamictest; bridges.

1. Introduction

Structures suffer from age effects, climate and often-growing traffic (for the case ofbridges). All in-service structures require some form of maintenance for monitor-ing their integrity and condition to prolong their service life. To ensure structuralsafety, structural health monitoring (SHM) has emerged as a reliable, efficient andeconomical approach to monitoring structural performance, detecting damage andmaking corresponding maintenance decisions. Some SHM techniques are based on astructure’s dynamic property variations (frequencies and mode shapes) before andafter damage.1 For most existing structures, the initial dynamic test has never beenrealized, and therefore reference dynamic properties are not available. The afore-mentioned inconvenience leads to the development of methods that can be directlyapplied to a structure’s in situ response.

Most of these methods are based on Fourier transforms, which break down asignal into constituents of different frequencies. The Fourier transforms convert

11

12 A. Alvandi et al.

the signal from a time-based or space-based domain to a frequency-based domain.Unfortunately, in the transformation to the frequency domain, the time or spaceinformation is lost and it is impossible to determine when or where a particularevent took place. Fourier transforms should thus be used for nonstationary signals,when the interest lies in what spectral components exist in the signal, and not atwhich locations these occur.

To overcome these difficulties, the short-time Fourier transform (STFT) wasproposed.2 This windowing technique analyzes only a small portion of the signal ata time or space. In the STFT, the signal is divided into small-enough segments, sothese segments of the signals can be assumed to be stationary. For this purpose, awindow function, w, is chosen in such a way that its width is equal to the stationarysegments of the signal. The disadvantage of this procedure is that the precision ofthe time or space and frequency is governed by a unique window size for the entiresignal. Therefore, a more flexible method making use of multiple window sizes wouldbe required to determine with greater precision any particular features of a signalin time or space as a function of frequency.

The wavelet transform offers this flexibility and is therefore perceived as a newway to analyze signals that overcomes drawbacks that other signal processing tech-niques exhibit. The wavelet analysis is done in a similar way to the STFT analysis,in the sense that the signal is multiplied by a function similar to the window func-tion defined in the STFT, and the transform is computed separately for differentsegments of the time domain signal.

Recently, there have been intense research activities in the application ofwavelets in various fields of science and engineering, including time–frequency analy-sis, system identification and damage detection.3,4 Early studies, primarily devotedto fault diagnosis in machinery, have shown the capability of the wavelet trans-forms to detect abnormal transient signals generated by the damaged gears.5,6

Based on these preliminary results, wavelet analysis has been extended to damagedetection of structural components. For example, based on the wavelet analysisof a cracked and a noncracked beam, Tian et al.7 identified crack locations byrecognizing the arrival time of waves exhibiting different velocities. Yam et al.8 pre-sented an integrated method for damage detection of composite structures basedon their dynamic responses and combining the wavelet approach and the artifi-cial neural networks. Furthermore, Sung et al.9 showed that damage in compositelaminates that were indistinguishable in conventional analysis of acoustic emissionwaves could be decomposed and identified by the wavelet transform. For beam dam-age detection, the wavelet transform applied to the static displacement responsesor vibration mode shapes is most commonly used.10 Liew and Wang11 were amongthe first to apply the wavelet transform to the numerical mode shapes of a single-cracked beam for crack detection and demonstrated that the maximum variationsin the spatial wavelet transform along the beam’s length corresponded to the cracklocation. Hong et al.12 have provided further insight into the existing correlationbetween damage and the maximum variation in the wavelet transform based on

Bridge Integrity Assessment by Continuous Wavelet Transforms 13

mode shapes. Their results have been confirmed by Douka et al.,13 who have appliedthe wavelet transform to mode shapes of cracked cantilever beams. Furthermore,Chang and Chen14 applied the wavelet transform to the mode shapes of a multi-cracked beam for crack determination and used the experimental natural frequenciesto predict crack depths. Chang and Chen have also pointed out that the wavelettransform may exhibit maximum variation at the beam ends, even if no damagedsections were located near the beam extremities. To counteract this problem, Gen-tile and Messina15 have proposed eliminating boundary local maxima by applyingthe wavelet transform on the windowed vibration mode shapes, which decay to zeroat the beam ends. Obviously, in this manner, the wavelet transform is not able todetect any damaged sections located near the beam extremities (if they existed).

As modal parameters, notably frequencies and mode shapes, are a function of thephysical properties of the structure (mass, stiffness); any variation in the physicalproperties will result in variation of the structure’s modal properties. For this reasonand because of the fact that mode shapes may reveal a lot of information abouta structure’s global behavior, methods based on dynamic responses have becomepreferable to those based on static responses. Nevertheless, some contributions maybe found in the literature in this regard, such as the one by Wang and Deng16

and Quek et al.17 showing that damage-related maximum variations occur in thewavelet transform of the numerical displacement of beams with vertical embeddedcracks featuring different orientations and widths. Zue and Law18 showed that beamdamage can be located using the wavelet transform of the operational displacementtime history when the beam structure is subjected to the action of a moving load.

Bayissa et al.19 presented the implementation of wavelet analysis on experimen-tal data of a bridge structure. By using modal data of the I-40 Bridge subjected tovarious damage scenarios, the robustness of the technique for “real-world” applica-tion was verified. Grabowska et al.20 conducted experimental tests on an aluminumbar by adding different masses to the rod. According to these results, the maximumdifference between the measured and calculated values of wavelet decompositioncoefficients and energy rates is only 6% of the value range. Ren and Sun21 presenteda wavelet entropy-based structural damage identification method. By applying thisapproach to a 1:3 scaled bridge with different damage scenarios, they confirmedthe efficiency of the proposed techniques for damage identification. Li et al.,22 forthe identification of structural response variation, presented a method based on thecombination of empirical mode decomposition (EMD) and wavelet analysis. Thistechnique has been applied to experimental laboratory mode shapes of a beam anda plate. According to this study, the continuous wavelet transform (CWT) is moresuitable than the discrete wavelet transform (DWT) for damage detection and themodulus and gradient of the adopted two-dimensional wavelet transform are goodindices of the damage localization.

As mentioned in the previous paragraph, some issues of wavelet transforms fordamage assessment of beams have already been discussed in the scientific literature.Nevertheless, in the literature, there is little contribution regarding the usage of


wavelets for bridge damage assessment. In this context, this paper aims to studythe potential of CWTs for structural health monitoring of existing bridges. As it isthe first time that this technique is being used for bridge integrity assessment, beforeits direct application to experimental data, the authors believe that some featuresof this technique still have to be discussed through a numerical model of a bridge.For this purpose, after reviewing the theory behind this technique, the performanceand applicability of the technique will be investigated through its application to thenumerical model of a beam and an existing concrete bridge found in the province ofQuebec in Canada. Once these results are presented, the experimental in situ modalparameter of the bridge will be used for its integrity evaluation. The influence ofthe CWT and its parameters — transformation, scale and vanishing moment —will be studied. The efficiency of static and dynamic responses as input data willbe investigated and compared. Other factors, like the number of used mode shapesand the effect of the damage quantity and type on detection results, will be studied.In order to automate the evaluation procedure, all the aforementioned parameterswere introduced in a toolbox that allows them to be easily modified. Based on theresults, a discussion on the effectiveness of the CWTs for structural assessment ofexisting bridges will follow.

2. Continuous Wavelet Transforms

The CWTs are defined as the product of an analyzed signal x(t) and the basicwavelet function ψ∗

τ,s(t) as follows:

CWTψx (τ, s) =

∫x(t) · ψ∗

τ,s(t)dt, (1)

where the basic wavelet function is defined as

ψ∗τ,s(t) =

1√sψ

(t − τ

s

). (2)

As is seen in the above equation, the transformed signal is a function of two vari-ables, τ and s, which are respectively called the transformation and the scale para-meter. ψ∗

τ,s is the complex conjugate of the wavelet, and it is generally called themother wavelet. The term “mother” implies that the wavelet functions are derivedfrom one main function. All the wavelet functions (windows that are used) are thedilated or compressed and shifted versions of the mother wavelet. The transforma-tion term τ is related to the location of the window along the original signal thatcovers the time or space length in the series (Fig. 1). For more refined or generalanalysis, the τ step can be modified to a smaller or larger value.

The scale parameter s, which is defined as 1/frequency (period), is similar tothe scale in maps; high scales correspond to a nondetailed global view of the signal,and low scales correspond to a detailed view (Fig. 2). Both τ and s are parametersused to modify scale and transformation axes.

The CWT is the sum over all time or space of the signal multiplied by a scaledand shifted version of a mother wavelet. The results of transforms are wavelet


Fig. 1. Transformation step throughout the signal.

Fig. 2. Schematic wavelet function at scales (a) s = 1, (b) s = 1.5, (c) s = 2.

coefficients that show how well a wavelet function correlates with the analyzedsignal. If a signal and the wavelet function match closely, the resulting waveletcoefficient will be high. A high negative value for a wavelet coefficient denotes aclose match, but in an inverse direction, meaning that the shape of the wavelet isthe mirror image of the original signal. More details about the CWT can be foundin Refs. 23–25.

For better comprehension, Fig. 3(a) presents schematically wavelet coefficientsobtained from a wavelet function applied to a signal. The figure illustrates well

(a) (b)

Fig. 3. Wavelet coefficients: (a) for different scales and transformations; (b) in a bar graph for agiven scale.


the dependence of the wavelet coefficients on the corresponding scales and trans-formations. In this study, for a better demonstration of the influence of scales (s)on wavelet coefficients, the wavelet coefficients for a given scale (s) are presentedin two-dimensional graphs. As an example, Fig. 3(b) presents one of these resultswhere the wavelet coefficients have been normalized and are shown in bars. In orderto be able to cover all the measurement points, a unit step for the transformationwas used (∆τ = 1). In this way, a transformation axis can be replaced by the mea-surement point axis. It is important to note that the transformation variable is ∆τ

and not τ itself; ∆τ changes the wavelet interval on the measurement points. Usinga unit interval, all measurement points are scanned and multiplied by the waveletfunction.

3. Wavelet Transform Properties

The two important wavelet properties are admissibility and regularity, which willbe explained in the following paragraphs.

3.1. Admissibility

It can be shown that the square integrable wavelet function ψ∗τ,s(t) must satisfy the

admissibility condition26: ∫ |ψ(ω)|2|ω| dω < +∞, (3)

where ψ(ω) stands for the Fourier transform of ψ∗τ,s(t). The admissibility condi-

tion implies that the Fourier transform of the wavelet ψ∗τ,s(t) vanishes at the zero

frequency:

|ψ(ω)|2ω=0 = 0. (4)

This zero value at the zero frequency also means that the average value of thewavelet in the time domain must be zero:∫

ψ∗τ,s(t)dt = 0, (5)

i.e. the wavelet function must be oscillatory. In other words, ψ∗τ,s(t) must be a wave.

3.2. Regularity condition

The time bandwidth product of the wavelet transform is the square of the inputsignal, and for most practical applications this is not a desired property. Thereforeone imposes some additional conditions on a wavelet function in order to makethe wavelet transform decrease rapidly with a decreasing scale s. These are theregularity conditions and they state that the wavelet functions should have somesmoothness and concentration in both time and frequency domains. In the following


paragraph, we will explain the regularity condition using the vanishing momentconcept.

Let us consider the following Taylor series at x = 0:

f(x) = f(0) +f ′(0)

1!x +

f ′′(0)2!

x2 +f ′′′(0)

3!x3 + · · · + fn(0)

n!xn. (6)

Using the presented Taylor series, Eq. (1) can be rewritten as

CWTψx (0, s) =

1√s

[x(0)

∫ψ

(t

s

)dt +

x′(0)1!

x

∫ψ

(t

s

)dt

+x′′(0)

2!x2

∫ψ

(t

s

)dt +

x′′′(0)3!

x3

∫ψ

(t

s

)dt + · · ·

]. (7)

Considering the wavelet moment as Mp:

Mp =∫

xpψ(t)dt, (8)

and substituting Eq. (8) in Eq. (7), the latter can be rewritten as follows:

CWTψx (0, s) =

1√s

[x(0)M0s +

x′(0)1!

M1s2 +

x′′(0)2!

M2s3

+ · · · + xn(0)n!

Mnsn+1 + · · ·], (9)

where p varies from 0 to n, and xp stands for the pth derivative of x. From theadmissibility condition we already know that the zeroth moment equals zero asM0 = 0 [Eq. (5)]. Therefore, the first term of Eq. (9) equals zero. If we manageto make other moments up to Mn equal to zero as well, the CWT coefficientsCWTψ

x (0, s) will decay as fast as sn+2. In this way, Eq. (9) can be written for adifferent number of wavelet moments, which is known in the literature as a numberof vanishing moments.27

As the smoothness and concentration of the wavelet functions depend on thenumber of vanishing moments, the wavelet functions are defined and identified bytheir numbers of vanishing moments.23,28 In other words, the wavelet transform canbe considered as the nth derivative of the signal x(t) smoothed by a wavelet moment(M) at the scale s. If a signal has a singularity at a certain point x, meaning that itis not differentiable at x, then the wavelet transform coefficients will have relativelylarge values at this point.29 By using a large scale, the convolution of the signalx(t) with M removes small signal fluctuations, and consequently detection of onlylarge signal variations is possible.30 As an example, Fig. 4 illustrates a Gaussianwavelet function found in Matlab Toolbox,31 where the digits 2, 4 and 6 representthe number of vanishing moments.


Fig. 4. Wavelet functions: (a) gauss 2, (b) gauss 4, (c) gauss 6.

4. Wavelet Transform Selection

The selection of an appropriate wavelet transform and the choice of its vanishingmoments are also key elements for effective use of the wavelet analysis in dam-age detection. Hong et al.12 proved that for beam crack detection the number ofvanishing moments should be at least 2. As the wavelet coefficient represents thecorrelation between the wavelet and the spatial signal, in the absence of damagethe best wavelet will be the one that exhibits a zero (or near-zero) wavelet coeffi-cient value. In other words, wavelets that are more similar to the signal shape willpresent better results.

For the signals used in this study (mode shape and deflection) based on varioussimulations, it was concluded that wavelets with four vanishing moments betterguarantee the zero values at undamaged areas. Obviously, for structural responsesthat are similar to the polynomial of order higher than 4, the use of wavelets with ahigher number of vanishing moments is necessary. Based on simulation on numeri-cal and experimental data obtained from the most commonly used wavelets, it wasconcluded that the Symlet (Sym) and Gaussian (Gauss) wavelets with four vanish-ing moments were the most powerful wavelets in detecting structural singularitieswhile using mode shape and/or deflection. The advantage of Gaussian wavelets hasalso been discussed by others.4,15

5. Damage Assessment Using CWTs

The main idea supporting the use of wavelets for structural assessment is basedon the fact that structural modifications introduce discontinuities in a structureresponse.32 The measured mode shape or deflection of a structure can be treated asa spatially distributed signal. Once a signal or its derivatives is available, it can beanalyzed through the wavelet transforms. A sudden change or peak in the analyzedwavelet coefficient may indicate the location of the damage or possible structuralmodification.

Spatial data is ideally recorded on the entire length of the structure and, forstructures with a significant width, it can be recorded according to more than onemeasuring line. In this study, in order to compare the efficiency of the dynamic


and static responses, the first three mode shapes and the static deflections of astructure have been used for damage assessment purposes. In order to automatethe evaluation procedure and be able to verify the influence of the wavelet function,the number of vanishing moments and the scale (s) in the assessment procedure, theCWT toolbox was developed.

6. CWT Toolbox

The CWT toolbox was created to automate the damage assessment procedure.A variety of wavelet functions with different vanishing moments were introducedin the toolbox. As seen in Fig. 3, the resulting wavelet coefficient depends on thewavelet function, scale (s) and transformation step (∆τ) parameters. With thehelp of the CWT toolbox interface, the wavelet function and the two parameterscan easily be modified and their influence on the detection results evaluated. Theprogram can be applied to one or two sets of recorded data, respectively presentedhere as damage identification through damaged and undamaged data, and damageidentification through damaged data. The following paragraphs give more detailsabout the CWT toolbox.

6.1. Damage identification through damaged data

In practice, the in situ response of existing bridges can be registered, but seldomis a previous set of records available for comparison purposes (reference state). Insuch cases, damage assessment should be based on one data set representing thecurrent state of the structure, called in this study the damaged data. One of thekey characteristics of the CWT in contrast to similar existing methods1 is that itcan make an evaluation using the damaged data only. In this case, the maximalamplitude of the wavelet coefficient localizes the modified or damaged zones. In thisstudy this wavelet coefficient has been normalized regarding its maximum value.

6.2. Damage identification through damaged and undamaged data

By using the CWT toolbox, the CWT can also be applied to two data sets usu-ally recorded at two different times in a structure’s life, called in this study theundamaged and damaged data. Figure 5 shows the program user interface withthe resulting wavelet coefficients at about 30 different measurement points alonga structure length, herein a simply supported beam. One can see that the dam-age area has been located near the 8th measurement point. For a given scale (s)and unit transformation step (∆τ = 1), the detection results are shown on a two-dimensional graph. In Fig. 5, the two wavelet coefficient graphs (a healthy state andan unknown damaged state) are being compared and the wavelet coefficients aregiven in the y-axis. As these graphs are obtained from the same wavelet functionswith identical scales (s) and transformation step (∆τ),their maximum variation


Fig. 5. Interface of the continuous wavelet transform toolbox.

is assumed to be the quantity of damage in the localized zone. In this study,the variation of the wavelet coefficient before and after damage has been calcu-lated and normalized to its maximum value and presented as a wavelet coefficientvariation.

6.3. Important issues of the CWT toolbox

6.3.1. Scale parameter

One important issue with the wavelet transform techniques is the choice of theappropriate scale (s). Results show that the selection depends on data qualityand damage type. As the structure assessment procedure is precisely conducted toidentify damage type and localization, it is obvious that the selection of the scale’sparameters may represent a challenge. Recall that inappropriate scale selection willresult in inaccurate damage detection. In order to counteract this problem, theCWT toolbox presents the possibility of determining the wavelet coefficient for theentire sum of the defined scale. It is particularly useful when the damage and/orquality of the recorded data are not known. In this case, the damage will be scannedthrough all possible scales.


6.3.2. False detection at extremities

The CWT is defined as the integration of the product of a wavelet and the signal ofinfinite length, i.e. x ∈ 〈−∞, +∞〉. Since the mode shape or deflection of a structuretreated as a spatially distributed signal has a finite length, i.e. x ∈ 〈0, L〉, a borderdistortion problem appears. Owing to this discontinuity at structure extremities,the wavelet coefficients achieve an extremely high value.33 As previously mentioned,one of the outstanding features of the CWT is the possibility of its direct applicationto the damaged data. Hence, false detections, which are sometimes more importantthan the damage itself, are often observed at the boundaries. The influence ofboundary effects can be reduced by extending the signal beyond the boundary.It is obvious that the length of the extended signal depends on the scale of theused wavelet function. There are different ways to extend the signal: extension byzeros, by reflection, by periodicity and by extrapolation.34,35 Extension by zerosassumes that the signal is zero outside the domain; this method creates artificialdiscontinuities at the border. Another method is signal extension by reflection,which assumes recovery of the signal outside its original support by symmetricboundary value replication. Reflection generally introduces discontinuities in thefirst derivative at the border. It is also possible to recover the signal by its periodicextension. This method also creates discontinuities at the border.

In this study, after examining all presented potential methods, it was concludedthat the spline extrapolation based on three neighboring points36 provides betterresults. For example, a beam deflection line signal with 20 measurement points isused for extrapolation purposes: see Fig. 6. This figure compares linear, cubic andspline extrapolation with ten extrapolated measurement points at each extremity.

Fig. 6. Extrapolation applied to a beam deflection response: ◦ linear extrapolation, ∗cubic extrapolation, spline extrapolation.


As shown, the linear and cubic extrapolations are closer to the initial curve. How-ever, due to its curvature, the first and second derivatives of the spline are con-tinuous and therefore provide better results. As artificial extensions at extremitieswill be taken off at the end of the procedure, they do not have any influence onthe final detection results. The toolbox allows the number of extension points tobe easily modified and their influence on results to be evaluated. This goes to showthat when two data sets are used and the wavelet coefficients difference is calcu-lated, the repeated false detections at extremities vanish, though the extrapolationprocedure is not needed.

7. Application of CWTs to Structures

In the following section, the performance of direct application of the CWTs toone data set (damaged data) and of application to two data sets (undamaged anddamaged data) will be studied through two practical examples. Here, all detectionresults were obtained from Gauss 4 with a unit transformation step (∆τ = 1), andscale 2. It should be noted that all wavelet coefficients obtained from direct appli-cation of CWTs to the damaged data were extrapolated by the spline extrapolationwith 10 extending measurement points at each extremity.

7.1. Simply supported beam

The feasibility and performance of the CWTs are first examined via a numeri-cal example representing the structure of a simply supported beam (Fig. 7). Thebeam was equally divided into 30 two-dimensional plane 42 elements in the ANSYSenvironment.37 All elements were assumed to be made of the same material, withthe following characteristics: modulus of elasticity E = 32GPa, mass densityρ = 2400kg/m3 and Poisson ratio υ = 0.25. The beam was subjected to a staticload of 1.5 kN at midspan. For the dynamic behavior, the beam’s free vibration wasconsidered. The vibration mode shapes and deflections were evaluated at 31 nodesalong the beam length, as shown in Fig. 8.

Fig. 7. Numerical model of a simply supported beam.


Fig. 8. Mode shapes and deflections of the simply supported beam: ◦ first mode shape,∗ second mode shape, � third mode shape, static deflection.

7.1.1. Damage scenarios

To show the potential of the CWTs for damage assessment, the structure wassubjected to two damage scenarios by reduction of the rigidity (EI): (1) in the8th element and (2) in both the 8th and the 22nd element. In order to examinethe damage quantity effect on the detection results, for both damage scenario, therigidity of the selected elements was reduced to 20%, 10% and 7% of their originalvalues (Fig. 9). Since visual inspection of the damaged and undamaged mode shapesdoes not permit to distinguishing one from the other (Fig. 8), the damaged modeshapes are not illustrated here. Next, the CWT toolbox will first be applied directlyto the damaged data and then to both the damaged and the undamaged data.

7.1.2. Damage identification through damaged data

For damage identification through damaged data, the CWT toolbox is applieddirectly to the mode shapes and deflections of the beam after each damage sce-nario. Figure 10 presents the detection results for the first damage scenario (20%

Fig. 9. Introducing damage into the beam elements.


Fig. 10. Detection results for the first damage scenario (20% damage in element 8): (a) first modeshape, (b) second mode shape, (c) third mode shape.

damage in element 8). As seen in this figure, for the first two mode shapes, thewavelet coefficient shows some structural modifications at or near the damaged ele-ment. However, such modifications are not seen in the third mode shape (Fig. 10).Figure 11 presents the detection results for the second damage scenario, with 20%damage in elements 8 and 22. According to this figure, some modifications are seenin the wavelet coefficient at or near the damaged elements (elements 8 and 22).However, in contrast to the first two mode shapes, the third does not show impor-tant sensitivity to such rigidity reductions. It should be noted that for the firstand second damage scenarios with 10% damage the same results were obtained.Regarding static response, for both damaged scenarios important detections wereseen at the load location, which were more important than the damage itself. Here,these results are not presented, but further results in this regard will be presentedin the next subsection (concrete bridge results).

Fig. 11. Detection results for the second damage scenario (20% damage in elements 8 and 22):(a) first mode shape, (b) second mode shape, (c) third mode shape.


7.1.3. Damage identification through damaged and undamaged data

Figure 12 presents wavelet coefficient variations for the first damage scenario (20%and 10% damage in element 8) based on the first mode shape. According to thisfigure, for both damage intensities (20% and 10%) the predefined damaged elementwas localized. As results obtained from the second and third mode shapes showedsimilar trends, they are not presented here. For the same damage scenario, Fig. 13presents the detection results based on static responses. In this case, for both of thedamage intensities 10% and 7%, some false detections were seen in the central partof the beam. In this case, 10% damage in element 8 was successfully localized, but,in reducing the damage to 7%, the maximum wavelet variation was not located atdamaged element 8.

In order to verify the potential of the wavelet transform for simultaneous dam-age, the second damage scenario (rigidity reduction in elements 8 and 22) was stud-ied. Figure 14 presents the detection results using the first mode shape regarding

(a) (b)

Fig. 12. Detection results for the first damage scenario (damage in element 8, based on the firstmode shape): (a) 20% damage, (b) 10% damage.

(a) (b)

Fig. 13. Detection results for the first damage scenario (damage in element 8, based on the staticresponse): (a) 10% damage, (b) 7% damage.


(a) (b)

Fig. 14. Detection results for the second damage scenario (damage in elements 8 and 22, basedon the first mode shape): (a) 10% damage, (b) 7% damage.

(a) (b)

Fig. 15. Detection results for the second damage scenario (10% damage in elements 8 and 22):(a) second mode shape, (b) third mode shape.

two predefined damage intensities (10% and 7%). As seen in this figure, for bothdamage intensities the two damaged elements were successfully identified. Figure 15,for the same damage scenario, presents the results of the second and third modeshapes, and once more the damaged elements were localized. Simultaneous damageassessment was also performed through static responses. Results for two damageintensities (10% and 7%) are shown in Fig. 16. It can be seen that, in the caseof 10% damage, the two damaged elements were localized; nevertheless, when thedamage was reduced to 7%, one of the damaged elements (element 22) was betterlocalized than the other (element 8). The above results tend to confirm the higherperformance of dynamic responses for simultaneous damage compared to staticresponses.

7.1.4. Multiscale results

In this study, all results were obtained using scale 2. In order to better understandthe influence of scale on detection results, Fig. 17 presents the beam results for


(a) (b)

Fig. 16. Detection results for the second damage scenario (damage in elements 8 and 22, basedon the static response): (a) 10% damage, (b) 7% damage.

(a) (b)

Fig. 17. Detection results for the second damage scenario, using different scales based on thefirst mode shape: (a) 7% damage in element 8, (b) 7% damage in elements 8 and 22.

7% damage based on different scales. As seen in this figure, scale variation hasan important impact on detection results. For example, for both damage scenarios[Figs. 17(a) and 17(b)], an increasing scale causes some false detections at thebeam’s center. Nevertheless, the scale parameter is an important issue in the waveletanalysis and more investigation by the authors is in progress.

7.2. Concrete bridge

In this subsection, the performance of the wavelet transforms for damage assessmentof existing bridges will be examined. For this purpose, some damage scenarios wereintroduced into the numerical model of the studied bridge. Before presenting theresults, a brief description of the bridge is given below.

7.2.1. Bridge description

The studied bridge, built in the 1960s, consists of a prestressed concrete box overthree continuous spans with variable moment inertia (Fig. 18). It has an overall


Fig. 18. Elevation view of a three-span concrete bridge.

Fig. 19. Bridge geometry.

length of 161.2m, with a central span of 80.8m and side spans of 40.4m. Thegeometry of the bridge is shown in Fig. 19. A full description of this bridge can befound in Refs. 38 and 39.

7.2.2. Numerical model of the bridge

A finite element model of the bridge was developed with the help of the ANSYSfinite element program. The SHELL99 elements defined by eight nodes with sixdegrees of freedom at each node (translations in the nodal x-, y- and z-directionsand rotations about the nodal x-, y- and z-axes) were used.37 Here, the first threenumerical mode shapes (Fig. 20) and the static responses of the bridge, measuredin three measurement lines along the deck shown in Fig. 21, were used for evalua-tion purposes. In order to verify the effect of the static load location, the bridge’s


(a) (b) (c)

Fig. 20. Dynamic behavior of the bridge in flexion: (a) first mode shape, (b) second mode shape,(c) third mode shape.

Fig. 21. Three measurement lines defined on the deck.

(a) (b)

Fig. 22. Static load locations: (a) in the center; (b) in the right span.

deflection was calculated by applying a static load at two different positions alongthe centerline of the deck (Fig. 22).

7.2.3. Damage scenarios

Two damage scenarios were introduced into the numerical model: (1) element rigid-ity reduction in the deck’s middle part [Fig. 23(a)], (2) settlement of a column ofthe bridge [Fig. 23(b)]. In the first scenario, the rigidity of elements located atthe middle span was reduced to 50%, 20%, 5% and 2% of their original values[Fig. 23(a)]. In the second scenario, a settlement of 40 cm and 10 cm of a col-umn was simulated [Fig. 23(b)]. In both scenarios, the bridge response was reg-istered at 101 measurement points along the deck. As previously done for the


(a) (b)

Fig. 23. Damage scenarios: (a) rigidity reduction in the middle span, (b) settlement of a column.

beam, for example, the CWT toolbox will first be applied to the damaged dataand subsequently to both the damaged and the undamaged data.

7.2.4. Damage identification through damaged data

7.2.4.1. First damage scenario (damage in the deck’s middle part)

(i) Detection results based on dynamic dataFigure 24 presents the results obtained from the CWT toolbox applied directly tothe dynamic damaged data of the first damage scenario with a rate of 50% damage.According to this figure, by using the first three mode shapes the damaged zone wassuccessfully localized. However, in the case of the second mode shape some detec-tion was seen at the bridge’s northern extremity. In order to verify the efficiency ofthe technique for a reduced rate of deficiencies, in the same damage scenario thedamage rate was reduced to 5% and 2%. Figure 25 shows the results, which confirmthe successful localization of the damage with the first mode shape. Although notpresented here, by using the second and third mode shapes the damage was suc-cessfully localized. However, by using the second mode shape some detection wasseen at the bridge’s northern extremity that was already seen for the 50% dam-age (Fig. 24). In order to verify the effect of the number of measurement pointson the detection results, Fig. 26 illustrates the same results presented in Fig. 25,using 21 measurement points. It is to be noted that 21 measurement points was areal number of sensors used in the experimental in situ test. As the figure shows,by reducing the number of measurement points to 21 for both damage intensities(5% and 2% rigidity reduction), the damaged area was successfully localized. Theseresults confirm that the 21 measurement points used in the experimental test weresufficient for wavelet analysis. However, wavelet analysis is very sensitive to thenumber of measurement points, and increasing this number by interpolation mayhave an important influence on the detection results.

(ii)Detection results based on static responseFor the same damage scenario (50% damage in the deck’s middle part), in order toverify the efficiency of the static response for damage detection, a static load was


(a) (b)

(c)

Fig. 24. Detection results for the first damage scenario (50% damage): (a) first mode shape,(b) second mode shape, (c) third mode shape.

(a) (b)

Fig. 25. Detection results for the first damage scenario, using the first mode shape: (a) 5%damage, (b) 2% damage.

applied at two different positions along the bridge deck (Fig. 22). The correspond-ing detection results are shown in Fig. 27. This figure illustrates well the effectof the load location. In Fig. 27(a), as the damage and load were both located inthe deck’s middle part, their influence acts concurrently. In Fig. 27(b), due to the


(a) (b)

Fig. 26. Detection results for the first damage scenario with 21 measurement points, using thefirst mode shape: (a) 5% damage, (b) 2% damage.

(a) (b)

Fig. 27. Detection results for the first damage scenario, based on static response: (a) static forcein the center, (b) static force in the right span (x = 122 m).

static load, the distinct influence of the load and damage can be noticed. Althoughnot presented here, the same observations were made for a reduced rate of damage(10%). These results show the effectiveness of the wavelet transforms for damageassessment by their direct application to the damaged static responses. Neverthe-less, particular attention must be paid to the load location.

7.2.4.2. Second damage scenario (settlement)

(i)Detection results based on dynamic and static dataFigure 28 presents the detection results for both dynamic and static data of the sec-ond damage scenario, where a 10 cm settlement was introduced at the right column.As these results show, by using dynamic and static damaged data, the settlementzone (x = 122m) was successfully localized. For all three used mode shapes, somewavelet coefficient variation is observed in the deck’s middle part (80m). This phe-nomenon could be attributed to the structural behavior modifications caused by


(a) (b)

(c) (d)

Fig. 28. Detection results for the second damage scenario (settlement 10 cm), based on dynamicand static responses: (a) first mode shape, (b) second mode shape, (c) third mode shape, (d)deflection.

the settlement itself. In the case of the static response, the settlement zone was suc-cessfully localized [Fig. 28(d)]. However, prior to extrapolation the false detectionswere always observed at extremities. The conclusions for a 40 cm settlement werethe same as those for a 10 cm column settlement. These results are not presentedhere.

7.2.5. Damage identification through damaged and undamaged data

In this subsection, using the CWT toolbox, the wavelet transform will be appliedto both damaged and undamaged dynamic and static data, and the results will becompared and discussed.

7.2.5.1. First damage scenario (damage in the deck’s middle part)

(i)Detection results based on dynamic dataFigure 29 presents wavelet coefficient variations of the first damage scenario fora 50% damage rate. According to this figure, for first and second mode shapes


(a) (b)

Fig. 29. Detection results for the first damage scenario (50% damage, based on dynamic data):(a) first mode shape, (2) second mode shape.

(a) (b)

Fig. 30. Detection results for the first damage scenario, based on the first mode shape for lowdamage intensities: (a) 5% damage, (b) 2% damage.

the predefined damage zone (i.e. central portion of the middle span) was preciselylocalized. To verify the performance of the technique for low damage intensities,the damage rate was reduced to 5% and 2%. Figure 30 presents these results. Onecan see that the damage zone is well detected but its localization is defined withless precision compared with the results of Fig. 29.

(ii)Detection results based on static dataAs previously seen, the static load location has an impact on the detection results.Figure 31(a) presents the wavelet coefficients before and after damage in the centralmeasurement line. It can be seen from this figure that, before introducing damage,due to the static load some modifications are observed in the central part of thedeck. However, introducing damage in the center adds some more modifications tothis area [Fig. 31(a)]. Figure 31(b), for all three measurement lines shown in Fig. 21,presents the normalized wavelet coefficient variation. Although the damaged zone


(a) (b)

Fig. 31. Detection results based on static force in the deck’s center for 50% damage: (a) waveletcoefficient comparison before and after damage in the central measurement line; beforedamage, ◦ after damage; (b) wavelet coefficient variation throughout the deck.

(a) (b)

Fig. 32. Detection results based on the static load at x = 122 m: (a) wavelet coefficient com-parison in the central measurement line; before damage; ◦ after damage; (b) waveletcoefficient variation before and after settlement in three measurement lines throughout the deck.

is well localized, some modifications are seen at the extremities. As these modifi-cations at extremities are not exactly the same before and after damage, it can beconcluded that they are due to the structural modifications caused by the damageitself. Figure 32(a) presents the same results, where the static load was appliedat x = 122m. It illustrates well the influence of load location on both damagedand undamaged response at x = 122m. Nevertheless, for undamaged response,some modifications are seen in the deck’s central part, which are attributed to therigidity variation (thickness) of the deck in that area. As false detections at theextremities were exactly the same for both undamaged and damaged wavelet coef-ficients [Fig. 32(a)], they disappear while one is calculating the wavelet coefficientvariation [Fig. 32(b)].


7.2.5.2. Second damage scenario (settlement)

(i)Detection results based on dynamic dataFigure 33 presents the detection results using the first mode shape for the 10 cmsettlement that was simulated at the bridge’s right column. Although not presentedhere, similar results were obtained for the second and third mode shapes. In otherwords, the settlement zone was successfully localized; however, some importantdetections are seen at extremities. These modifications at extremities are due tothe introduced settlement, which has influence in these areas. As results for the50 cm settlement were similar, they are not presented.

(ii)Detection results based on static data (settlement 10 cm)Figure 34(a) compares two wavelet coefficients before and after settlement at thecentral measurement line, with the static load at the deck’s center. As this figure

Fig. 33. Detection results for the second damage scenario (10 cm settlement), based on the firstmode shape.

(a) (b)

Fig. 34. Detection results based on static force applied in the deck center: (a) comparison of thewavelet coefficients in the central measurement line; before damage, ◦ after damage;(b) comparison of wavelet coefficients in three measurement lines throughout the deck.


shows, owing to the settlement and static load, some structural modifications areseen in the damaged and undamaged response. Because of the settlement, the loadeffect and the modifications at the bridge extremities are not the same before andafter settlement. Figure 34(b) shows the final detection results based on waveletcoefficient variations in the three measurement lines along the deck. This figurelocalizes the settlement zone. However, as discussed in Fig. 34(a), some modifica-tions are also seen at the load position and at the extremities. Concerning the staticcharge, it is to be noted that the uniform load was replaced by the concentratedone and the same results were obtained.

8. Bridge Integrity Evaluation

Dynamic tests under ambient traffic were carried out in 2004 with the mobile labo-ratory of the Quebec Ministry of Transport. Eight accelerometers measuring verticalvibration were installed and fixed in a concrete caisson in two measurement linesalong the bridge.38 These two measurement lines, at deck extremities, are shownin Fig. 21. The first three identified mode shapes were used for bridge evaluationpurposes. Wavelet analysis based on experimental mode shapes was compared tothat based on the finite element mode shapes. Before applying wavelets to thesedata, experimental and numerical mode shapes were compared using modal assur-ance criteria (MAC)40; an average correlation of 95.61% was obtained. In fact, thisclose correlation confirms the performance of finite element mode shapes as refer-ence data for comparison purposes. Wavelet parameters applied to experimentaland numerical data are as follows:

• Wavelet: gauss 4.• Scale (s): 2; transformation step (∆τ = 1): 1.• Number of measurement points: 21.

Bridge evaluation results interpolated on the deck are presented in Fig. 35. Accord-ing to this figure, by using the first three mode shapes, the structural modificationswere detected in the central part of the deck. Furthermore, structural modifica-tions are located in the western part of the deck. Although not presented here, thisbehavior difference between two deck sides was not visible while comparing modeshapes.

9. Numerical Noise Simulation

In order to study the reliability of wavelet transforms regarding environmentalparameters called noise in this study, a numerical model of a continuous beamwas considered. This model consisted of 20 elements (2-node linear elements with 3degrees of freedom per node). For better demonstration these elements are presentedin two dimensions (Fig. 36). Values for material properties of beam elements wereassigned as follows: (1) elastic modulus E = 210 × 109 kg/m2; (2) linear mass


(a) (b)

(c)

Fig. 35. Bridge evaluation by wavelet analysis, using the first three mode shapes.

Fig. 36. Test model: random force applied on the simply supported beam.

density ρ = 798.1kg/m3. Values for geometric properties were: (1) cross-sectionalarea A = 0.02m2; moment of area I = 666.7×10−8 m4. Applying a random force atnode 6, a linear dynamic analysis was performed and the signals at each nodal pointwere measured. The analysis time was 15 s, with a sampling time of 0.002 s. Differentlevels of damage (1%, 2%, 4%, 6%, 8%, 10%, 15%, 20% and 30% rigidity reduction)


were introduced in element 5. After initial investigations, it was determined that anadditional 4% noise level could be considered as a high level of noise. In consequence,the following noise levels were added to signals: 0.1%, 0.25%, 0.5%, 1%, 1.5%, 2%,2.5%, 3% and 4%. For each noise simulation, the noise percentage was multipliedto the standard deviation value of the signal in each measuring channel, and itwas randomly added to the other component of the current channel (multiplying torandom numbers between −1 and 1). The noisy signals (before and after damage)were treated by the random decrement technique41 — a modal identification methoddeveloped in the Matlab c©42 environment. The triggering condition in this procedurewas a positive point and the sampling time was 0.002 s.

A total of 50 simulations were generated and 50 series of modal parametersbefore and after damage were obtained. By undertaking 50 noise simulations, thetotal probability results were stabilized. After signal treatment, the identified noisymodal parameters, before and after damage, were used as entrance data for waveletanalysis. The first three identified frequencies were 5.88, 23.52 and 52.2Hz. Onlythe first mode shape corresponding to the first frequency was considered.

Detection probabilities corresponded to different damage, and noise ratios arepresented in Fig. 37. In fact, these probabilities represent the number of timesthat wavelet analysis was able to detect the damaged element out of all 50 sim-ulations. These results illustrate clearly the dependence of detection probabilitieson the damage intensity and noise ratio. As seen in this figure, for low damageintensities (less than 10% rigidity reduction), the detection probabilities are influ-enced to some extent by noisy data. These results underline the importance ofenvironmental parameters for detection results, especially regarding low damageintensities. It must be mentioned that in this study, scale 2 was used. However, fornoisy data the scale parameter plays an important role and further investigationis needed.

Fig. 37. Detection probability for different noise/damage ratios.


10. Conclusions

Damage causes perturbations in the dynamic and/or static responses of a struc-ture; however, such perturbations may not always be directly detectable throughthe recorded responses. Applying the wavelet transform technique to the dynamicmode shapes and static deflections of a beam and a bridge with different dam-age intensities showed promising results in localizing these perturbations. For thebridge structure, damage scenarios similar to the practical ones were introducedinto its numerical model. The high sensitivity of the proposed technique to dam-age intensity and type suggests that it may prove to be a high performance toolfor structural health monitoring of existing bridges. Comparing wavelet coefficientsthrough experimental and numerical mode shapes revealed structural modificationareas, situated in the deck’s center.

An important outstanding feature of this technique that distinguishes it fromother existing structural health monitoring techniques1 is its damage assessmentcapability using only one data set. The damage detection technique based onwavelet analysis is in a very early stage of development. This study is the firstapproach to use this technique for damage assessment of a bridge structure. There-fore, more features of the technique should be examined. This approach has tobe applied to more complex and experimental data; in particular, its performanceregarding noisy field data must be investigated. Nevertheless, the current studyof wavelet analysis for damage assessment leads to the following conclusions andsuggestions:

(1) By applying the continuous wavelet transform toolbox to the static or dynamicresponses of a structure, its current state can be evaluated in a few minutes.

(2) For nonsignificant or simultaneous damage, owing to its higher resolutiondynamic responses usually provide better results than static ones. Neverthe-less, important modifications can be scanned through the static responses.

(3) Some damage can be scanned better through some mode shapes than throughothers. Therefore, in using dynamic data, at least a few mode shapes have tobe considered (Figs. 10 and 11).

(4) In using static response, particular attention must be paid to the load loca-tion; detections are always seen at this location. This becomes more critical fornonsignificant damage.

(5) For local or concentrated damage, for the same signal length, small scales pro-vide precise detection, but for nonlocalized damage (when damage is spreadthroughout the structure) or for noisy data, higher scales provide better detec-tion. However, when there is no knowledge of these parameters, in order not tomiss any detection, a wavelet coefficient for the entire sum of the defined scalesis recommended (this option is available in the continuous wavelet transformtoolbox).

(6) For local or concentrated damage, where damage does not have any influenceon global structural behavior, the comparison of two data sets (before and after


damage) provides more precise results than direct application of the techniqueto the damaged data.

(7) The extrapolation procedure at structure extremities has been proven to be ableto overcome the false detection at the boundaries. It is strongly recommendedwhen one is applying the continuous wavelet transforms directly to the damageddata.

(8) Using a few mode shapes and/or deflections, Symlet (Sym4) and Gaussian(Gauss4) wavelet functions with the four vanishing moments were recognizedas the best candidates for structural evaluation.

(9) The number of measurement points is a key element of wavelet analysis; asufficient number interpolated measurement points is needed. However, moreinvestigation in this regard is required.

Acknowledgment

The authors are grateful to the Quebec Ministry of Transportation and the Nat-ural Science and Engineering Research Council of Canada for their assistance andsupport.

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