STRUCTURAL CONTROL AND HEALTH MONITORINGStruct. Control Health Monit. 2014; 21:331–341Published online 30 April 2013 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/stc.1564
Nondestructive damage detection in Euler–Bernoulli beams usingnodal curvatures—Part II: Field measurements
Selcuk Dincal and Norris Stubbs*,†
Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, College Station, TX 77843, USA
The Interstate 40 (I-40) bridges located over Rio Grande in Albuquerque, NM were demolished in thesummer of 1993 and were replaced by a new bridge. Prior to the replacement, researchers from NewMexico State University and Los Alamos National Laboratory were able to introduce incrementallevels of damage on the bridge and performed experimental modal analyses at each damage stage.The results of the experiments were utilized to evaluate the performance of five previously publishednondestructive damage identification methods, which included the following: the damageindex method proposed by Stubbs et al. [1]; the mode shape curvature method proposed byPandey et al. [2]; the change in flexibility method proposed by Pandey and Biswas [3]; the changein uniform load surface curvature method proposed by Zhang and Aktan [4]; and the change instiffness method proposed by Zimmerman and Kaouk [5].
The damage index method utilizes the equivalency of the fraction of modal strain energy before andafter damage. For the ith mode, the fraction of modal strain energy in the jth element corresponds to themodal strain energy contained in the jth element of the beam divided by the modal strain energycontained in the entire beam. Equating the predamage and postdamage fractional modal strain energiesyields the damage indicator for the jth element of the beam at the ith mode.
*Correspondence to: Norris Stubbs, Zachry Department of Civil Engineering, Texas A&M University, 3136 TAMU, CollegeStation, TX 77843, USA.†E-mail: [email protected]
The mode shape curvature method utilizes the difference between the curvature mode shapes of theintact and damaged beams to detect damage. For a given magnitude of moment, the curvature is in-versely proportional to the flexural stiffness. Thus, as the stiffness of the damaged zone is reduced,the absolute difference between the curvature mode shapes of the intact and damaged beams increases.This difference can then be used to predict the location of the inflicted damage.
The change in flexibility method uses the difference between the predamage and postdamage modalflexibilities to detect structural damage. Using only a few lower modes of vibration, modal flexibilitymatrices before and after the inflicted damage may be estimated. For each measurement location, themaximum absolute value of the elements in the corresponding column of the flexibility differenceindicates the DOF where damage is located.
The change in uniform load surface curvature method utilizes certain characteristics of the changemode shape curvature and the change in flexibility methods. The sum of the unit load flexibilities (thesum of all columns of the flexibility matrix) represents the deformed shape of the structure loaded at allthe DOFs. This deformed shape is referred to as the uniform load surface. The absolute change in thecurvatures of the uniform load surfaces before and after damage is an indicator of the damage location.
The change in stiffness method uses the predamage and postdamage stiffness matrices to detectdamage. The stiffness matrices of the structure are approximated from the modal parameters measuredbefore and after damage. The damage vector for the ith mode is defined as the product of the differencebetween the estimated predamage and postdamage stiffness matrices and the ith damaged mass-normalized mode shape vector.
Farrar and Jauregui [6] utilized the five aforementioned nondestructive evaluation (NDE) method-ologies to perform nondestructive damage identification on the I-40 bridge at each incremental levelof damage. The objective of this paper is to evaluate the performance of the NDE methodologypresented in the accompanying paper ‘Nondestructive damage detection in Euler–Bernoulli beamsusing nodal curvatures – Part I: Theory and numerical verification’ using field measurements fromthe I-40 bridge study. The experimental modal data utilized in this paper correspond to the refinedset of sensor measurements collected from the midspan of the north girder via forced vibration tests.Experimental results include the predamage and postdamage translational mode shapes andeigenfrequencies of the structure. A detailed description of the structure, the proposed damagescenarios, and the modal data collected after each incremental level of damage are available in theliterature [6–8]. Only a brief overview of the experiment is provided here.
2. DESCRIPTION OF THE STRUCTURE
The I-40 bridge consisted of twin spans for each traffic direction. Each span was made up of a concretedeck, which was supported by two welded steel plate girders and three steel stringers. Each bridge wasmade up of three identical sections. The sections were structurally independent. A single section hadthree spans; the two end spans were of equal length with approximately 40m (131 ft), and the centerspan was approximately 50m (163 ft) long. A detailed description of the structure is given in [9].The elevation view and a typical cross-section of the bridge are given in Figures 1 and 2, respectively.
3. INFLICTED DAMAGE SCENARIOS
Four incremental levels of damage were inflicted at the midspan of the north plate girder of the bridgeby making various torch cuts in the web and flange of the girder. These damage scenarios were
Figure 1. Elevation view of the instrumented section of the bridge (adapted from [6]).
Figure 2. Typical cross-section of the bridge (adapted from [6]).
NDE USING NODAL CURVATURES—PART II: FIELD MEASUREMENTS 333
intended to simulate fatigue cracking in the girder. The first damage scenario was a 61 cm (2 ft) long,1 cm (3/8 in) wide web cut centered vertically at the midheight of the 305 cm (120 in) thick plate girder.In the second damage scenario, the initial cut was extended to include the bottom of the web. In thethird damage scenario, in addition to the 182.9 cm (6 ft) web, the bottom flange was cut halfway infrom either sides below the web. Finally, the 182.9 cm web and full bottom flange were removed inthe fourth damage scenario. Figure 3 depicts the torch cuts made at various portions of the north girderfor simulating the sequential damage.
Figure 3. Damage scenarios inflicted on the north plate girder E-1: damage scenario 1, E-2: damage scenario 2,E-3: damage scenario 3, and E-4: damage scenario 4.
4. RESPONSE PARAMETERS UTILIZED FOR DAMAGE DETECTION
Forced vibration tests were performed on the undamaged bridge. These tests were repeated after eachincremental damage scenario had been introduced. Three sets of modal data were used for damagedetection. Experimental results included the predamage and postdamage translational mode shapesand eigenfrequencies of the structure. SET 1 represented the experimental modal data obtained fromthe cross-spectra analysis utilizing the refined set of sensor measurements collected from the midspanof the north girder. SET 2 represented the experimental modal data obtained via global polynomialcurve fit of a coarser set of sensor layout. SET 3 corresponded to the numerically generated modal data.SET 1 modal data are utilized for damage evaluation in this paper. The input signal was not monitoredin this data set, and vertical accelerations were measured at the midheight of the plate girder. Figure 4shows the locations of the accelerometers within the midspan of the north girder. The first two bendingmodes of SET 1 modal data for the undamaged structure and for four sequentially damaged structuresare shown in Figure 5. Resonant frequencies for these modes are listed in Table I. These modal datawere reported in [6]. Note that a detailed inspection of the frequency results presented in Table Ireveals that data reported for damage scenarios 1 and 2 may be questionable, because theory andexperience dictate that frequency should decrease with the inflicted damage. It should be notedhowever that frequency measurements are extremely sensitive to temperature effects and/ormeasurement errors and noise.
Figure 4. Sensor layout for the north plate girder (adapted from [6]).
334 S. DINCAL AND N. STUBBS
5. RESULTS OF THE PREVIOUS DAMAGE DETECTION STUDIES
Damage detection results of the five previously mentioned NDE methodologies utilizing SET 1 modaldata are summarized in Table II. These level II (damage localization only) results were generated byFarrar and Jauregui [6]. The authors observed that the location of the most severe damage scenario(E-4) was identified by all methods. However, excluding the damage index method, the aforemen-tioned methodologies provided ambiguous damage localization results for the other less severe damagescenarios. The damage index method outperformed all other NDE algorithms by identifying the truelocation of damage in all damage cases. However, the study provided no information pertaining todamage severity estimates.
●, damage located; ●●, damage narrowed down to two locations; ●●●, damage narrowed down to three locations; ○, damagenot located.
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6. NDE OF THE I-40 BRIDGE USING THE PROPOSED METHODOLOGY
6.1. Summary of the proposed damage evaluation algorithm
This section provides a synopsis of the NDE methodology, which was previously presented in theaccompanying paper ‘Nondestructive damage detection in Euler–Bernoulli beams using nodal curva-tures – Part I: Theory and numerical verification’. Assuming that the internal stress resultants in thebeam were not affected by the inflicted damage, the following system of equations were proposedfor damage evaluation
A NN�2ð Þ� NEð Þ b NE�1ð Þ ¼ B NN�2ð Þ�1 (1)
where the NE� 1 (i.e., NE by 1) vector, b, denotes the unknown damage ratio vector. The (NN� 2)� (NE) matrix A contains the curvatures for the undamaged beam, and the (NN� 2) � 1 vector B
contains the curvatures for the damaged beam. Solution to Eq. (1) was given by
b ¼ A�1P B (2)
where A�1P is the pseudo-inverse of A. SVD is used to compute the generalized (Moore–Penrose
pseudo-inverse) of matrix A. The predicted severity of localized damage was expressed in terms of thedamage indicator bj (i.e., for the jth damaged element) as
ajP ¼ ΔEIjEIj
¼ EIj� � EIjEIj
¼ 1bj
� 1 (3)
6.2. Basic measurements required by the theory
The proposed damage evaluation methodology utilizes the predamage and postdamage point curva-tures to localize and quantify damage in beam-type structures. Point curvatures along the beam lengthcan be estimated using the transverse displacement profiles. Thus, the first step in damage prediction isto obtain the deformed shapes of the undamaged and damaged beam. Note that, the pristine and thedamaged beam must be subjected to the same external load to satisfy the equality of the internal stressresultants before and after damage. The modal flexibility matrix serves as an excellent tool to achievethis end, because each row of the modal flexibility matrix can be interpreted as the deformed shape of astructure because of a unit load applied at the corresponding DOF. Furthermore, the low-frequencydominance of the modal flexibility provides a good estimation of the static displacement profile byusing only few of the lower modes of vibration [10]. Utilizing the transverse DOFs of the first two
bending modes and the corresponding natural frequencies, modal flexibilities of the I-40 bridge areapproximated as follows:
uj ¼Xr
i¼1
fji
limifi (4)
where mi, fi, and li denote the ith modal mass, mode shape, and eigenvalue, respectively. The jthcomponent of the ith modal vector is denoted by fji. The jth modal flexibility vector, uj, represents theapproximated static deflection profile due to a unit load applied at the jth DOF [10]. It should be notedthat mass-normalized mode shapes are necessary to accurately construct the modal flexibility matrix.Because the SET 1 modal data were extracted using output-only modal analysis techniques (the inputsignal was not monitored), mass-normalized modal amplitudes were approximated using an assumedidentity matrix [6]. The exact scaling of the modal flexibility is not expected to affect the damageprediction results [6,11]. Therefore, the ith modal mass, mi, in Eq. (4) is computed as
mi ¼ fTi Mfi (5)
where M is assumed to be the identity matrix.
6.3. Damage prediction results for the I-40 bridge damage scenarios
Excluding the two support nodes, 11 measurement points lead to a set of nine modal flexibility vectors(each corresponding to the deformed shape of the beam due to the unit load applied at a different sensorlocation). Because the sensor interval is not uniform, cubic spline interpolation with uniform 15.2 cm(6 in) intervals is used to generate a even and finer sensor layout along the length the bridge. Thisprocess leads to 323 nodal deflection points (NN=323) including the two nodes that correspond to thesupports. The predamage and postdamage curvatures can be estimated from these nodal deflections byusing the central difference formula. Thus, nine sets of point curvatures are available for the undamagedand damaged structure. The system of linear equations given in Eq. (1) must hold for each of thesecurvature sets. An optimal solution to the system may be obtained by using the entire set of pointcurvatures concurrently. Therefore, the 322� 1 damage index vector (given in Eq. (2)) can solved byusing the 2889� 322 (9� (NN� 2)� (NE)) curvature matrix A and the 2889� 1 (9� (NN� 2) � 1)curvature vector B.
Damage localization results are reported in terms of the element damage ratios (Eq. (2)). The sever-ity of the predicted damage is computed using Eq. (3). Note that the proposed damage scenariosreduced the moment of inertia of the north plate girder. Therefore, in this paper, the true damageseverity is quantified by dividing the difference between the baseline and damaged moment of inertiasto the baseline inertia. Let aT represent the true damage severity inflicted at the beam girder by meansof the torch cuts. Then, for a cut section, aT can be computed as
aT ¼ ΔII0
¼ I � I0I0
(6)
where I0 represents the moment of inertia of the intact section and I corresponds to the reducedmoment of inertia due to damage. Note that I is computed with respect to the updated neutral axis aftereach incremental level of damage.
6.3.1. Damage scenario 1. Figures 6 and 7 depict the damage prediction results for damage scenario 1. Inthis damage scenario, the bending stiffness of the girder is estimated to be reduced by 0.2% (using Eq. (6)).Table III summarizes the performance of the damage evaluation methodology for the damage scenario.
6.3.2. Damage scenario 2. Figures 8 and 9 depict the damage prediction results for damage scenario 2.In this damage scenario, the bending stiffness of the girder is estimated to be reduced by 12% (usingEq. (6)). Table IV summarizes the performance of the damage evaluation methodology for thedamage scenario.
6.3.3. Damage scenario 3. Figures 10 and 11 depict the damage prediction results for damagescenario 3. In this damage scenario, the bending stiffness of the girder is estimated to be reduced by
Figure 6. Damage localization result for damage scenario 1.
Figure 7. Damage severity estimate for damage scenario 1.
Table III. Assessment of the damage prediction accuracy for damage scenario 1.
Damage location (m)
Error (%)
Damage severity (%)
Error (%)True Predicted True Predicted
25.1 24.9 0.4 �0.2 �35.2 35.0
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44% (using Eq. (6)). Table V summarizes the performance of the damage evaluation methodology forthe damage scenario.
6.3.4. Damage scenario 4. Figures 12 and 13 depict the damage prediction results for damagescenario 4. In this damage scenario, the bending stiffness of the girder is estimated to be reduced by96.3% (using Eq. (6)). Table VI summarizes the performance of the damage evaluationmethodology for the damage scenario.
6.4. Discussion of results
With the results of the case studies, the following observations can be made regarding the performanceof the proposed NDE methodology under field conditions.
Damage localization accuracy was quantified by a dimensionless position error given in the accom-panying paper. The damaged localization error, eL, was as low as 0.4% for damage scenario 1. Damagescenarios 2–4 consistently indicated a localization error of 1.6%. In the view of the case studiesperformed, the proposed methodology performed equally as well as the damage index method for dam-age localization and better than all of the other previously mentioned NDE algorithms according to theresults given by Farrar and Jauregui [6].
The error in damage severity estimation was quantified by utilizing the fractional error in stiffnessprediction. The predicted damage severity corresponds to the peak value of damage severity computeddirectly from Eq. (2). The true damage severity inflicted at the beam was computed using Eq. (6).Excluding damage scenario 1, the proposed NDE methodology provided satisfactory severity estimateswith the dictated sensor layout. The predicted damage severity exceeded the severity of the inflicteddamages in damage scenarios 1 and 2. Experience suggests that a relatively coarse sensor layout
Figure 10. Damage localization result for damage scenario 3.
Figure 11. Damage severity estimate for damage scenario 3.
Table V. Assessment of the damage prediction accuracy for damage scenario 3.
Damage location (m)
Error (%)
Damage severity (%)
Error (%)True Predicted True Predicted
25.1 24.3 1.6 �44.0 �29.7 �14.3
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usually provides lower estimates for damage severities (as in the cases of damage scenarios 3 and 4),and enhancing the senor resolution should lead to better severity predictions. Damage evaluationresults for damage scenarios 1 and 2 do not to support this observation. Note however that modalanalysis results showed an increase in the frequency content with incremental levels of damage fordamage scenarios 1 and 2. This error may be due to temperature effects and/or measurement errorsand noise. For this reason, the reliability of the NDE results, which assumed constant environmentalconditions for these two damage scenarios, is debatable. The estimated damage severities appearedto be reasonable for damage scenarios 3 and 4. Note that the prescribed torch cuts caused significantstiffness reductions in these damage scenarios, which might have violated the assumption of constantinternal force distribution before and after damage. While the apparent stiffness loss for damagescenario 4 turned out to be quite large, the true and the predicted damage severity estimates are in goodagreement (�96.3% and �88.0% fractional loss, respectively).
The objective of this paper was to evaluate the performance of the NDE methodology presented in theaccompanying paper ‘Nondestructive damage detection in Euler–Bernoulli beams using nodal curva-tures – Part I: Theory and numerical verification’ by using field measurements. Experimental modaldata collected from the I-40 bridge were utilized to accomplish the stated objective. The point curva-tures required for damage evaluation were estimated from the transverse displacements, which hadbeen approximated from the modal flexibility matrix. Damage was successfully localized in all damagescenarios of the I-40 bridge experiment. The method provided satisfactory damage severity estimatesfor damage scenarios 3 and 4. Deviations from the inflicted (true) damage severities for damagescenarios 1 and 2 were linked to possible variations in environmental conditions at the time of testingor measurement errors. Overall, considering that the damage index method was the only NDE
NDE USING NODAL CURVATURES—PART II: FIELD MEASUREMENTS 341
methodology, which identified the true damage location in the Los Alamos study, it is noteworthy thatthe proposed NDE method accurately predicted the location of damage for each damage scenario andadditionally provided realistic estimates of damage severity by using the same modal information.Experience suggests that a finer sensor arrangement within the localized damage region might haveimproved the estimates of damage severity.
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