Structural damage diagnosis using high resolution images

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  • Structural damage diagnosis using high resolution images

    Gongkang Fua,*, Adil G. Moosab

    aDepartment of Civil and Environmental Engineering, Wayne State University, Detroit, MI 48202, USAbDiversied Computer Engineering and Development, Clawson, MI 48017, USA

    Abstract

    Structural damage diagnosis is a critical element for structural safety monitoring. This paper presents anew approach to diagnosing structural damage using high-resolution images by CCD camera. For mini-mum eort of instrumentation, this approach oers an unprecedentedly large amount of spatially intensivedata to be used for diagnosis. A probabilistic data processing procedure is presented here for diagnosingstructural condition. The proposed method is applied in the laboratory for a bridge model using a highresolution CCD camera for data acquisition. Results show that this method can clearly identify both theexistence and the location of all the structural damages used. The smallest of them introduced a 3% loss tothe structures stiness. # 2002 Published by Elsevier Science Ltd.

    Keywords: Damage diagnosis; High-resolution images; Image processing; Probabilistic diagnosis

    1. Introduction

    Many civil structures need periodic examination for proper operation and/or for maximizinglife span. Such monitoring is now practiced mainly through general visual inspection, followed bylocal in-depth examination when needed. These two steps are referred to as global and localdiagnosis, respectively. A typical application of this type of monitoring is to bridge structures inthe transportation infrastructure system.Global diagnosis for large structures is often labor intensive, costly, and possibly subjective.

    Signicant research eort has been made to develop physical testing techniques to improve, sup-plement, and/or even replace visual inspection. While nondestructive testing techniques for localdiagnosis for bridges have been noticeably advanced [1], techniques for global diagnosis have notreached a stage of routine application [2].The main reasons for this current state of the art can be summarized as follows: (1) previously

    proposed nondestructive testing methods for globally diagnosing bridges have been overwhelmingly

    0167-4730/02/$ - see front matter # 2002 Published by Elsevier Science Ltd.PI I : S0167-4730(02 )00004-8

    Structural Safety 23 (2001) 281295

    www.elsevier.com/locate/strusafe

    * Corresponding author. Fax: +1-313-577-3881.

    E-mail address: gfu@ce.eng.wayne.edu (G. Fu).

  • based on dynamic modal testing using accelerometers [3]. These sensors can perform measure-ment at a limited number of points on the structure. Covering a large area using many accel-erometers could be very expensive, and no techniques are available for eective diagnosis usingmeasurements from only a few points [4]. (2) The eectiveness of using modal testing to diagnose(including locating) small damage needs to be established and quantied. This is becausemeasurement data inevitably contain noise. This noise makes it dicult to identify the signal,especially when local and small damages are concerned. (3) Some approaches proposed to alsoinclude numerical modeling as a dominant or complementary tool (e.g., nite element modeling).Such modeling can also be costly for a large number of bridges in a transportation network. Inaddition, it remains unknown whether numerical modeling can realistically cover the incon-sistency between the nominally specied and the as-built conditions. Examples of such incon-sistencies include real concrete strengths being dierent from the specied values, non-structuralmembers adding structural stiness, etc.This paper presents a new approach to global diagnosis using high resolution images, which can

    be provided by a charge coupled device (CCD or CCD camera). A typical CCD is shown in Fig. 1,which has a sensor to receive light signals to process to digital images. The sensor here has anarray of 2048 by 2048 sub-sensors referred to as pixels. CCD has the following advantages inacquiring needed data for damage diagnosis for structural safety monitoring: (1) no sensors needto be attached to the structure, which signicantly reduces instrumentation costs including that

    Fig. 1. Apogee AP10 CCD with 50 mm lens.

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  • for interrupted normal operation of the structure. (2) A large number of points on the structurecan be covered for which measurement data are to be obtained. This oers an unprecedentedamount of spatially intensive data for enhancing diagnosis resolution and eectiveness. (3) Theprices for CCD have become very aordable and are expected to be more so in near future, dueto rapid advancements in the computer and electronics industry. Note also that, depending onoperation costs, the camera may be kept stationary at the site or brought to the site wheneverneeded.This paper rst presents the concept of using spatially intensive data for global diagnosis. It is

    followed by a discussion on measuring displacement using CCD images. Then laboratoryexperiments are presented for illustrating the proposed approach.

    2. Probabilistic advancing cross-diagnosis (PAC) method

    Due to the large amount of data that can possibly be made available, the proposed method usesa probabilistic framework for diagnosis including multiple structural signatures. In addition, ithas a strategy of focusing on neighborhoods of dierent size on the structure to advance theresolution of diagnosis. The details of this proposed method are discussed here.Assume that a number of, say H, sets of repeated measurement data are made available for the

    intact state of the structure to be monitored, denoted as [B]1L to [B]HL standing for the before state:

    B 1L B1L1 ; B1L2 ; B1L3 ; B1L4 ; . . . ; B1LJ

    . . . . . .B HL BHL1 ; BHL2 ; BHL3 ; BHL4 ; . . . ; BHLJ

    1

    The square brackets [ ] with a bolded letter inside are used here to indicate a matrix consistingof several vectors denoted by the bolded letters without brackets. Each vector BhLj (h=1, 2, . . .,H, j=1, 2,. . ., J) contains M data elements from a pre-selected data grid covering the interestedportion of the structure (possibly the entire structure). Accordingly, each of [B]1L to [B]HL con-sists of J vectors representing the replicates, i.e., J is the number of measurement replicates.Superscript L indicates the feature (i.e., a physical quantity) that is referred to. These featuresshould be inherent to the structure, responsive to structural damage, and otherwise invariant. Forexample, the following features are used in the experiments discussed below: D=displacement,S=slope, C=curvature, and C2=curvature squared. When the load on the structure remainsunchanged (e.g., the self-weight), these features are considered to be the structures inherentproperties. Note that other features can also be included if warranted. The HJ replicates in Eq.(1) are needed to deal with possible noise always present in measured data. Further, assume that anew measurement data set [A]L is made available for a structural state (an after state) to bediagnosed:

    A L AL1 ; AL2 ; AL3 ; AL4 ; . . . ; ALJ 2

    For the purpose of damage diagnosis, when a local area of the structure is focused, not neces-sarily all M data points are needed at the same time. Say, only N (4M) points are used, which

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  • are from a neighborhood on the structure. Note that N can be changed to focus on other neigh-borhoods of dierent geometric size and/or location. This neighborhood is referred to as N-neighborhood hereafter. As a typical situation, consider an N-neighborhood data set taken as asubset of the entire data set dened in Eqs. (1) and (2), denoted as [B]hL,N (h=1,2,. . .,H) and[A]L,N:

    B 1L;N B1L;N1 ; B1L;N2 ; B1L;N3 ; B1L;N4 ; . . . ; B1L;NJ

    . . . . . .

    B HL;N BHL;N1 ; BHL;N2 ; BHL;N3 ; BHL;N4 ; . . . ; BHL;NJ 3

    A L;N AL;N1 ; AL;N2 ; AL;N3 ; AL;N4 ; . . . ; AL;NJ 4

    This typical subset of data is used to diagnose the N-neighborhoods structural condition, asdiscussed below.

    2.1. Correlation coecient

    Correlation coecient (CC) is used here to indicate how much two vectors are alike to eachother. It is dened as the normalized dot product of the two data vectors X and Y from the sameN data points:

    CCXY XY i1;:::;NXiYi

    j1;:::;NX2j

    1=2

    k1;:::;NY2k 1=2 5

    where Xi and Yi are the data elements of the two vectors: (X1, X2, X3, . . ., XN)=X and (Y1, Y2,Y3, . . ., YN)=Y. CC for the N-neighborhood is calculated referring to a reference vector B

    L,Navg as

    follows:

    CCIL;NBi BIL;Ni BL;Navg ; :::::;CCHL;NBi BHL;Ni BL;NavgCCL;NAk AL;Nk BL;Navg i; k 1; 2; . . . ; J 6

    BL;Navg i1;2;...;JB1L;Ni j1;2;:::;JB2L;Nj . . .k1;2;...;JBHL;Nk

    = HJ 7

    These CC values range from 1 to +1 according to the denition in Eq. (5). For the prob-abilistic analyses discussed below, it is desirable that CC be converted through the Fisher transfer[5] to the following variables r. As a result, they range from negative to positive innity, and arecloser to being normally distributed:

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  • r1L;NBi 0:5Ln 1 CC1L;NBi

    = 1 CC1L;NBi

    ; . . . . . . rHL;NBi 0:5Ln 1 CCHL;NBi

    = 1 CCHL;NBi

    rL;NAk 0:5Ln 1 CCL;NAk

    = 1 CCL;NAk

    i; k 1; 2; :::; J 8

    These data elements are re-organized as vectors, to be used below for diagnosing changes due tostructural damage:

    r1L;NB r1L;NB1 ; r1L;NB2 ; r1L;NB3 ; :::::::::; r1L;NBJ

    ::::::

    rHL;NB rH;NB1 ; rHL;NB2 ; rHL;NB3 ; :::::::::; rHL;NBJ

    rL;NA rL;NA1 ; rL;NA2 ; rL;NA3 ; ::::::::; rL;NAJ

    9

    These H+1 r vectors contain information on how much alike between the reference vector BL,Navgand each vector in BhL,Ni and A

    L,Nk (h=1, 2, ,..., H; i, k=1, 2, ...,J).

    2.2. Probabilistic damage indicator

    For each feature (e.g., D, S, C, or C2 in the application experiments presented below), a like-lihood ratio (LR) for structural damage is dened as follows:

    LRL;N LFAL;N;B1L;N&:::&BHL;N h1;:::;HLFBhl;N;B1L;N&:::&BHL;N

    =H 10

    where

    LFAL;N;B1L;N&:::&BHL;N LnPrL;NA

    ;r1L;NB

    &:::&rHL;NB

    Pr1L;NB

    &:::&rHL;NB

    ;rL;NA

    LFB1L;N;B1L;N&:::&BHL;N LnPr1L;NB

    ;r1L;NB

    &:::&rHL;NB

    Pr1L;NB

    &:::&rHL;NB

    ;r1L;NB

    ::::::LFBHL;N;B1L;N&:::&BHL;N LnPrHL;N

    B;r1L;NB

    &:::&rHL;NB

    Pr1L;NB

    &:::&rHL;NB

    ;rHL;NB

    :11

    Pr1, r2 in Eq. (11) is the probability that one data set r1 belongs to the other data sets (r2s)probability density function, dened as:

    Pr1;r2 maxr1min r2

    f2 r dr

    maxr1 the maximum element in data set r1minr1 the minimum element in data set r1 12

    f2(r) Is the probability density for data set r2. It can be assumed to be a normal distribution,unless another distribution is deemed to be more appropriate. The statistical parameters such asthe mean and standard deviation for f2(r) can be estimated using data set r2. Note that r1 and r2 in

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  • Eq. (12) are interchangeable. LRL,N in Eq. (10) indicates the likelihood that [A]L,N belongs to the

    population or family of [B]1L,N to [B]HL,N for feature L. It is expected to be low for a damagestate represented by [A]L,N because the mean values of the measured quantities will change (fromthose of [B]1L,N to [B]HL,N), to say the least. The negative term in Eq. (10) represents a benchmarkfor comparison, accounting for noise in the baseline data [B]1L,N to [B]HL,N. When [A]L,N is closeto or belongs to the [B]hL,N (h=1,2,. . .,H) population, LRL,N is close to zero, indicating extremelylow probability of damage. Accordingly, lower LRL,N values indicate higher likelihood ofdamage.

    2.3. Cross diagnosis using multiple features

    These likelihood factors in Eq. (10) for all the features are combined here into a comprehensivelikelihood factor (CLF) as follows to be used as a single index for easy and reliable cross diag-nosis:

    Comprehensive likelihood factor CLFN LwL LRL;N 13

    LwL 1; wL > 0 14

    CLFN is a weighed likelihood factor using all the features, indicating the likelihood of struc-tural damage. A lower CLFN indicates higher likelihood of damage for the data point repre-senting the N-neighborhood. wL Is the weight for feature L. It should be higher when thesensitivity of that feature to the damage of concern is higher. In other words, more sensitive fea-tures (e.g., slope and curvature as opposed to displacement) should have higher weights. Notethat this sensitivity is aected by the dependence between the feature and the damage, and is alsoinuenced by the data noise because noise makes it dicult to identify a signal. Note also thatthis approach is intended to be applicable when dierent features L are used.For a selected N, CLFN in Eq. (13) serves as a damage indicator for a physical point on the

    structure representing the N-neighborhood. Thus, CLFN plotted for all the N-neighborhoodsover the entire data grid is referred to here as a condition map of the structure. These conditionmaps for dierent N should be easy to understand for engineers performing the diagnosis. Someexamples of condition map are shown in Figs. 69 where low CLF values indicate the damagelocations. The experiments leading to these results are discussed later in detail.Also note that when N is large to cover a large area, CLFN may become insensitive to local

    damage. It is because only a local area may be aected by the damage and shows abnormalbehavior. CC calculation using Eqs. (5) and (6) has an averaging or diluting eect over theN-neighborhood. With N varying from a large to a small value, the resolution of diagnosis isincreased because the diluting eect becomes lower. When N becomes small enough (forexample N=2) the focus will be on the damages immediate vicinity. Note that when N=1, theCC calculation in Eq. (6) will not be needed, then the before and after data in Eqs. (3) and(4) will be directly compared in Eq. (10) for diagnosis. Namely, BhL,1i and A

    L,1k (h=1, 2, . . .,H; i,

    k=1, 2, . . ., J) in Eqs. (3) and (4) each reduces from a vector to a scalar (a data element). As aresult, Eq. (9) will be:

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  • Fig. 2. (a) A steel I-beam W615 imaged using CCD. (Area at midspan enlarged in Fig. 2b.) (b) Close view of mid-span section of I-beam in Fig. 2a. (One-pixel column of beam edge digitized in Fig. 2c.) (c) Intensity distribution invertical direction at an edge point in Fig. 2b. (Circled points=digitized intensity from image; solid curve=6th orderpolynomial tting).

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  • r1L;1B B1L;11 ;B1L;12 ;BIL;13 ; ::::::;B1L;1J

    :::::::rHL;1B BHL;11 ;BHL;12 ;BHL;13 ; ::::::;BHL;1J

    rL;1A AL;11 ;AL;12 ;AL;13 ; ::::::;AL;1J

    15

    Note again that the vectors in the parentheses are understood as scalars as explained. With thisunderstanding, the rest of data processing from Eq. (10) to (14) remains unchanged.

    Fig. 3. Data grid and simulated damage cases (stiness change severity given in Table 1).

    Table 1Testing program for damage diagnosis

    Test ID Description of damage introduceda Stiness loss (%)

    DC1 1 cm long cut to bottom ange (12 cm from midspan) 3.0

    DC2 3 cm long cut to bottom ange (12 cm from midspan at the same location of DC1) 9.5DC3 2 cm long cut to bottom ange (30 cm from midspan) 6.2DC4 4 cm long cut to bottom ange (30 cm from midspan at the same location of DC3) 13.0

    a Cut width: 4.0 mm.

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  • 3. Displacement identication using CCD images

    Measuring displacement using CCD essentially is a task of identifying the point of interest inthe images taken before and after its displacement. For structural displacement in a sub-milli-meters range or smaller, a sub-pixel resolution is usually required. This can be accomplished by,essentially, reversing the digitization process for images as follows.Identifying a point in an image is to identify the points unique light reection in the sur-

    rounding environment. Since monochrome images are used here, light reection is quantied bylight intensity recorded in the image. Accordingly, the required sub-pixel resolution can beachieved through the following two general steps. (1) Develop a continuous function of lightintensity by curve-tting the digitized intensities in the area of the interested point. This is torestore the continuous intensity that has been discretized or digitized. (2) Identify the location of

    Fig. 4. Denitions of N-neighborhoods. Note: X indicates the point representing the neighborhood, i.e., at whichCLFN is plotted.

    Fig. 5. 1-cm Cut in bottom ange for test DC1.

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  • Fig. 6. Condition map for test DC1 (1-cm cut to bottom ange, 3.0% stiness change between Points 55 and 56): (a)N=9; (b) N=5; (c) N=3; (d) N=1.

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  • the interested point as one satisfying an appropriate mathematical condition on the continuousintensity function obtained in the rst step.As an example, for the I-beams bottom edge in Fig. 2a, a polynomial was used for a one-

    dimensional curve tting in the vertical direction. This serves as Step 1, generally dened above.Then the inection point was used as the point of interest, dened as where the second derivativeis equal to zero. This condition is based on the fact that, in Fig. 2a, the edge is at a location wherelight intensity rapidly changes from very bright to very dark. Figs. 2b and 2c show the physicallocation and numerical curve tting for one pixel-column in the image of Fig. 2a. Note that thisapproach has been calibrated against measurements of dial gages with a resolution of 0.00254mm [6].

    Fig. 7. Condition map for test DC2 (3-cm cut to bottom ange, 9.5% stiness change between Points 55 and 56,N=1).

    Fig. 8. Condition map for test DC3 (2-cm cut to bottom ange, 6.2% stiness change between Points 139 and 140,N=1).

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  • Fig. 9. Condition map for test DC4 (4-cm cut to bottom ange, 13.0% stiness change between Points 139 and 140):(a) N=9; (b) N=5; (c) N=3; (d) N=1.

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  • 4. Laboratory application

    To demonstrate the proposed PAC method using CCD, laboratory experiments were con-ducted on a structure model. In these experiments, bridge structures were targeted for applica-tion. A simply supported steel W615 I-beam shown in Fig. 2a was used as a bridge structuremodel. These tests provided the beams deection data for the before and after states. Thefeatures included in the CLF calculation in Eq. (13) are: D=displacement, S=slope, C=curva-ture, and C2=curvature squared. A noise reduction algorithm based on Spokoiny [7] was appliedto the raw data to enhance their quality. This also covered the slope and curvature data numeri-cally obtained using the displacements.The Apogee AP10 monochrome CCD in Fig. 1 was used in these experiments for data acqui-

    sition. It has an array of 20482048 pixels, a 14-bit digitizing system, and a pixel size of 14 m. TheCCD is controlled by a Pentium-II computer. The lens also shown in Fig. 1 is an ordinary 50-mmlens for convenient focusing and minimum distortion. The CCD sensor is very sensitive to light sothat no additional lighting was needed in the normally illuminated laboratory.Each pixel of the CCD images can be used as a data point in the data grid. In order to have a

    manageable number of data points for this rst trial, one of every 11 pixels along the beam lengthwas used, resulting in a 173-point grid (i.e., M=173) as shown in Fig. 3. Of course, the gridresolution can be further improved by using more pixels. Fig. 4 shows the denitions of N-neighborhoods used here. The central point in an N-neighborhood was used as the representativepoint for the neighborhood. Namely, the CLF value for that neighborhood is plotted at the cen-tral point marked as X in the condition maps.As mentioned above, the beams deection, slope, curvature, and curvature squared [i.e., L=D,

    S, C, and C2 in Eqs. (1)(15)] were used in this application. The following weights for Eq. (14)were used, based on a computer simulation for the experimental program [2,6]:

    wD 0:1; wS 0:2; wC 0:3; wC2 0:4 16

    Table 1 shows four cases of simulated damage to the model structure included in the test pro-gram. These cases represent local damages and were introduced by cutting the steel beam using ahand held electrical grinder. Fig. 3 indicates the locations of cut in the beam. For these tests,H=2 and J=4 were used for acquiring replicates. For future eld application in monitoringbridge structure over a long term, more replicates (i.e., larger H and J) perhaps will be neededbecause more variation in the image data would be expected compared with the laboratory con-dition.Test DC1 had a 1-cm long and 4-mm wide cut between Points 55 and 56, 12 cm away from the

    mid-span section. This cut is shown in Fig. 5, which introduced a 3.0% stiness reduction to thecross section. Fig. 6 exhibits the condition maps for this damage case for four selections of N-neighborhood with N=9, 5, 3, and 1. The N-neighborhood denitions are given in Fig. 4. Thesemaps demonstrate a typical process of advancing diagnosis from larger N-neighborhoods tosmaller ones. It is seen that when N is relatively large (9 and 5) the damage location is not veryclear, although the area between Points 50 to 57 shows a little lower CLF values indicating ahigher likelihood of damage. When N becomes smaller (N=3 and 1 in Fig. 6c and d), the damage

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  • location is much clearer. Fig. 6d shows that the vicinity of damage can be identied as betweenPoints 54 to 57, because of the observed lower CLF there. These points indicate three intervals ofthe grid, or a length of 30.5 cm=1.5 cm along the beam length.Test DC2 had a cut at the same location as Test DC1 but the size was increased from 1 to 3 cm,

    as indicated in Table 1. It introduced a 9.5% reduction to the cross section stiness. Fig. 7 dis-plays the condition map for this case for N=1. Other selections of N-neighborhood have CLFshowing very similar behavior as in Fig. 6. They are not shown here because of space limitation.A stronger signal can be clearly seen in the condition maps in Fig. 7, compared with that shownin Fig. 6d, both with N=1. This comparison also contrasts the damage severity in these twocases, indicating a more severe damage than DC1. The lowest CLF values are seen at Points 54and 55, actually indicating the damages immediate vicinity. Again, note that the real distance is 5mm on the I-beam between these two points.After Tests DC1 and DC2, another cut of 2 cm4 mm was made at a dierent location 30 cm

    to the right from the mid-span section. The cut was made between Points 139 and 140 in the datagrid, as shown in Fig. 4. This cut was made in an area with smaller deection, to observe diag-nosis eectiveness of the PAC method associated with the CCD used under that condition. Notethat if noise in deection is uniformly distributed over the beam, damages in low deection areasmay become more dicult to diagnose, because the signal (true deection) to noise ratio will besmaller.Test DC3 introduced a 2 cm4 mm cut, representing a 6.2% stiness reduction in addition to

    that of DC1 and DC2. Its condition map is shown in Fig. 8. Again, the low CLF values at andaround Point 139 clearly indicate a higher likelihood of damage in the vicinity. The lowest valueis at Point 139, which indeed is the damage location. Note that the condition map does notinclude Points 55 and 56 where the DC2 damage was.Increasing the cut in Test DC3 to 44 mm in Test DC4 introduced a 13.0% of stiness loss to

    the model structure. Fig. 9 shows the condition maps for this case, showing two damage areasover the entire grid from Points 1 to 173. As in Fig. 6 for damage case DC1, these CLF plotsconsistently point to the two damage areas. When N is large (9 and 5 in Fig. 9a and b) thedamage severity is shown relatively lower by higher CLF values but over larger surroundingareas. This is because of the diluting eect discussed earlier. When N becomes smaller, thedamage areas are narrowed down and the severity also becomes higher. Note that areas otherthan the two damage vicinities shows scattered low CLF values, but they do not consistentlypoint to an area for high likelihood of damage. The lowest CLF in these condition maps is shownat Point 140 and its immediate vicinity, which gives a very strong signal indicating the damagelocation.The condition maps for all these tests show not only the damage presence but also its location

    or vicinity. This represents a signicant advancement over the previously proposed modal testingmethods [8]. On the other hand, these condition maps show the damage-indicating signals spreadover an area not at a point. This may be caused by the grinder cut being not perfectly perpendi-cular to the longitudinal axis of the test beam, which could cause surrounding areas to be aec-ted. In addition, damage signals could attenuate to surrounding points in the noise reductionprocess [7]. In practical application, when positive global diagnosis as such is established, localdiagnosis can be called for to conrm and quantify the damage. It could be a local nondestructivetest focusing on the identied local area.

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  • These results show that CCD image data can be eectively used for diagnosing structural sti-ness loss. For eld applications, these images should include bottom surface of the primarymembers such as the slab (for slab bridges) and the beams (for multi-beam bridges). Since iden-tifying the edges or texture traces of these components can be performed as illustrated here to asub-pixel resolution, this method may be eciently used for bridge safety monitoring.

    5. Conclusions

    A probabilistic diagnosis method is developed here using high-resolution CCD for monitoringstructural safety. Laboratory experiments show that this method can detect structural damagewith 3% stiness reduction. These results indicate that eld experiment of this method iswarranted towards implementation.

    Acknowledgements

    It is gratefully acknowledged that the work reported here was funded by the US FederalHighway Administration. Thanks are also due to Drs. Steven Chase, Philip Yen, and Paul Fuchswith FHWA for their direction and assistance at various stages of the study. Dr. Haluk Aktanand many fellow graduate students of Wayne State University facilitated and assisted in thelaboratory experiments. Their contributions are appreciated.

    References

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    [2] Fu G, Moosa AG, Peng J. Probabilistic pattern recognition using coherent laser radar for bridge inspection. FinalReport to FHWA, USDOT, for Contract DTFH6197P00549, Department of Civil and Environmental Engi-neering, Wayne State University, November 1998.

    [3] Doebling SW, Farrar CR, Prime MB, Shevitz DW. Damage identication and health monitoring of structuraland mechanical systems from changes in their vibration characteristics: a literature review. LA-13070-MS Report,Los Alamos National Laboratory, May, 1996.

    [4] Fu G. Bridge Inspection: Modal Testing for Global Diagnosis. In: Shiraishi N, Shinozuka M, Wen JK, editors.7th International Conference on Structural Safety and Reliability. Kyoto, Japan: 2428 November, 1997. p. 475.

    [5] Sachs L. Applied statistics: a handbook of techniques, second ed. Springer-Verlag; 1984.[6] Moosa AG. Nondestructive damage detection for structures by optical methods and probabilistic diagnosis. PhD

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    [8] Alampalli S, Fu G, Dillon EW. Signal versus noise in fatigue damage detection by experimental modal analysis.ASCE Journal of Structural Engineering 1997;123(2):23745.

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