comac method damage detection

Engineering Structures 24 (2002) 501–515www.elsevier.com/locate/engstruct

Damage assessment in reinforced concrete beams usingeigenfrequencies and mode shape derivatives

J.-M. Ndambia,*, J. Vantommea, K. Harri b

a Royal Military Academy, Civil Engineering Department, Av. de la Renaissance 30, B-1000 Brussels, Belgiumb Royal Military Academy, Applied Mechanics Department, Av. de la Renaissance 30, B-1000 Brussels, Belgium

Received 22 May 2001; received in revised form 9 September 2001; accepted 21 September 2001

Abstract

The use of changes in dynamic system characteristics to detect damage has received considerable attention during the last years.This paper presents experimental results obtained within the framework of the development of a health monitoring system for civilengineering structures, based on the changes of dynamic characteristics. As a part of this research, reinforced concrete beams of 6meters length are subjected to progressing cracking introduced in different steps. The damaged sections are located in symmetricalor asymmetrical positions according to the beam tested. The damage assessment consists in relating the changes observed in thedynamic characteristics and the level of the crack damage introduced in the beams.

It appears from this analysis that eigenfrequencies are affected by accumulation of cracks in the beams and that their evolutionsare not influenced by the crack damage locations; they decrease with the crack damage accumulation. The MAC factors are lesssensitive to crack damage compared to eigenfrequencies, but give an indication of the symmetrical or asymmetrical nature of theinduced crack damage. Next to this, the COMAC factors, the strain energy evolution and the changes in flexibility matrices arealso examined as to their capability for detection and location of damage in the RC beams, the strain energy method appears tobe more precise than the others. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Damage; Dynamic system characteristics; Reinforced concrete; Eigenfrequencies; MAC; COMAC; Strain energy method; Flexibilitymatrix

1. Introduction

Nowadays, modal testing methods are widely used asnon-destructive tools in many engineering applicationsto detect and evaluate damage, including civil engineer-ing applications. This is due to the development of newpowerful systems in data acquisition and signal pro-cessing, allowing to determine accurately the dynamicsystem characteristics (DSCs). Nevertheless, the user hasto be careful because some non-controllable problemsrelated to measurement quality and accuracy may leadto wrong interpretation of the obtained measurements.Some of these problems are mentioned in Ref. 1. How-ever, the problem remains to establish a correct corre-lation between the changes observed in measured DSCs,

* Corresponding author. Tel.:+32-2737-6422; fax:+32-2737-6412.E-mail address: [email protected] (J..-M. Ndambi).

0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (01)00117-1

the damage appearance, the detectable severity and thelocation.

The basic idea of damage evaluation techniques basedon vibrations is that the dynamic characteristics are func-tions of the physical properties of the structure and there-fore any change in these properties caused by damageresults in the change of DSCs [2]. In concrete structures,it has already been proved that damage, either local orglobal, is associated with structural modifications, whichcan be observed through changes in dynamic character-istics: eigenfrequencies, modal damping ratios, modeshapes and derivatives [3,4]. This observation makes thedamage detection techniques based on changes of DSCspromising in civil engineering applications.

The strain energy method was applied by Cornwell etal. [5] to an aluminium plate to detect and locate damagein the plate using a saw to obtain localised damage. Theysuccessfully detected and located damage in an alu-minium plate using the damage index (see Section 2.2).

Pandey et al. [6] used the change in the flexibility

502 J.-M. Ndambi et al. / Engineering Structures 24 (2002) 501–515

matrices to detect damage in a (US standard) beamW12×16. The damage was simulated by opening boltsfrom the splice plates. Once again, the damage was welllocalised and this damage could be detected and local-ised by the change observed in the flexibility matricesof the structures.

This paper presents additional results obtained in theframework of a research program for evaluation ofdynamic techniques for damage detection in reinforcedconcrete (RC) beams. Increasing levels of crack damageare introduced by subjecting the beams to static loadswith increasing amplitude. After each step, dynamicmeasurements are performed to determine the dynamiccharacteristics. These characteristics are obtained by per-forming curve fit procedures on a number of measuredfrequency response functions (FRFs). The FrequencyDomain direct Parameter Identification (FDPI) techniqueimplemented on the CADA-X [7] system is used for thedynamic characteristics estimation. The universal filesproduced by the CADA-X system are imported to MAT-LAB software in order to calculate the strain energy dis-tribution and the flexibility matrices of the beams. Thecomparison of the set of measurements in damaged andundamaged state of the beams allows the sensitivityanalysis of eigenfrequencies, modal assurance criterion(MAC) factors, coordinate modal assurance to crackdamage in RC beams. The 1-D strain energy method isapplied to the beams to detect and locate the crack dam-age. For the last method, the damage index is used forthe damage detection.

2. Damage detection: theoretical aspects

2.1. Modal Assurance Criterion (MAC) andCoordinate Modal Assurance Criterion (COMAC)

Next to the basic modal parameters of structures suchas eigenfrequencies, modal damping ratios and modeshapes, some derived coefficients obtained from theseparameters can also be useful for damage detection instructures. The MAC and COMAC factors may be men-tioned in this category. These factors are derived frommode shapes and express the correlation between twomeasured mode shapes obtained from two sets of tests.The MAC and COMAC factors being linked to modeshapes, sufficient number of degrees of freedom (numberof measurement points) are needed to obtain accuratequantities. The next paragraph gives a short theoreticaldescription of the MAC and COMAC factors [8–10].

Let [�A] and [�B] be the first and second sets of mea-sured mode shapes in matrix form of sizes n×mA andn×mB respectively, with mA and mB the numbers ofmodes shapes considered in the respective sets and thenumber of measurement points. The MAC factor is thendefined for the modes shapes j and k as follows:

MAC(jk)�

��n

i�1

[�A]ji [�B]k

i �2

�n

i�1

([�A]ji)2 �n

i�1

([�B]ki )2

(1)

where j=1%mA and k=1%mB.[�A]j

i and [�B]ki are the ith components of the modes

[�A]j and [�B]k , respectively. MAC(j,k) factor indicatesthe degree of correlation between the jth mode of thefirst set A and the kth mode of the second set of modeshapes B. The MAC values vary from 0 to 1, with 0 forno correlation and 1 for full correlation. Therefore, if theeigenvectors [�A] and [�B] are identical, the corre-sponding MAC value will be close to 1, thus indicatingthe full correlation between the two modes. The devi-ation of these factors from 1 could be interpreted as adamage indicator in structures.

The COMAC factors are generally used to identifywhere the mode shapes of a structure from two sets ofmeasurements do not correlate. If the modal displace-ment in a coordinate i from two sets of measurementsare identical, the COMAC factor is close to 1 for thiscoordinate. A large deviation from unity can be inter-preted as damage indication in the structure. For thecoordinate i of a structure and using m mode shapes, theCOMAC factor is defined as follows:

COMAC(i)�

[�mj�1

�[�A]ji [�B]j

i �]2

�mj�1

([�A]ji)2 �m

i�1

([�B]ji)2

(2)

2.2. Strain energy method

The starting point of this method is the formulationof the strain energy of a Bernoulli–Euler beam given bythe relation (3). Two versions of this method exist: theone applicable to the beam-like structures (1-D) and theother one applicable to the plate-like structures (2-D).The next paragraph resumes some essential elements ofthe first method [5].

U�12�l

0

EI�∂2w∂x2�2

dx (3)

where U is the strain energy, EI is the flexural rigidityof the beam, l is the length of the beam and w is theshape function of the beam.

If a mode shape yi(x) is considered, the associatedstrain energy Ui is given by:

Ui�12�l

0

EI�∂2yi

∂x2 �2

dx (4)

503J.-M. Ndambi et al. / Engineering Structures 24 (2002) 501–515

Fig. 1. Static loading test: symmetrical and asymmetrical loadingconfigurations.

Table 1Static loading steps: symmetric configuration

Load steps Loads (kN)

ref 0st1 4st2 6st3 12st4 18st5 24st6 25.3

Considering a beam subdivided in N divisions, thestrain energy Uij associated with each sub-region j corre-sponding to the mode shape yi(x) is given by:

Table 2Static loading steps: asymmetric configurations

First set-up (a) Second set-up (b)

Load steps Loads (kN) Load steps Loads (kN)

ref 0 st1b 8st1a 7 st2b 10st2a 10 st3b 13st3a 13 st4b 19st4a 19 st5b 25st5a 25 st6b 35

st7b 53

Uij�12�aj+1

aj

(EI)j�∂2yi

∂x2 �2

dx (5)

with aj and aj+1 delimiting the sub-region j.From the strain energy related to each sub-region Uij

and the strain energy of the complete structure Ui,Cornwell et al. [5] define the fractional strain energy Fij

as follows:

Fij�Uij

Ui

(6)

leading to

�Nj�1

Fij�1 (7)

Considering the damaged structure, the quantitiesmentioned in Eqs. (4)–(7) can also be quantities obtainedfrom the damaged mode shapes yd of the structure.

From these quantities and after some mathematicalarrangements and making some assumptions, Eq. (8) canbe established [5].

(EI)k

(EI)dk

�

�ak+1

ak

�∂2ydi

∂x2 �2

dx

�l

0�∂2yd

i

∂x2 �2

dx

�ak+1

ak

�∂2yi

∂x2 �2

dx

�l

0�∂2yi

∂x2 �2

dx

�f d

ik

fik(8)

with (EI)k and (EI)dk corresponding to the flexural rigid-

ities of the sub-region k in the undamaged and damagedstates, respectively. In order to use all the measuredmodes shapes m in the calculation, the damage index bk

for the sub-region k is defined as:

bk�

�mi�1

f dik

�mi�1

fik

(9)

The value of the damage index b allows us to evaluatethe health state of the structure. bk expresses the degra-dation of the flexural rigidity in a certain zone of thestructure by the change in strain energy stored in thestructure when it deforms in its particular mode shape. InEq. (9), the mode shapes do not need to be normalized.

Assuming that the collection of the damage indices bk

represent a sample population of a normally distributed


Fig. 2. Dynamic measurements: measurement locations.

Fig. 3. Eigenfrequencies in function of static loading steps: sym-metrical configuration.

random variable, a normalised damage index zk isdefined as follows:

zk�bk−bk

sk

(10)

where bk and sk represent the mean and standard devi-ation of the damage indices, respectively.

2.3. Flexibility method

It has already been proved that the presence of cracksin a structure increases its flexibility. So, the experi-mental changes observed in the flexibility matrix can beinterpreted as a damage indication in the structure andmay allow one to evaluate and locate damage [11].

The principle of this method is based on the compari-

Fig. 4. Eigenfrequencies in function of static loading steps: asym-metrical configuration.

son of the flexibility matrices obtained from two sets ofexperimental mode shapes. The method is applicableonly if the mode shapes are mass-normalized to unity(fTMf=1). If this is the case, the measured flexibilitymatrices can then be estimated from the mass-nor-malized mode shapes. The following equations give therelationships between the system stiffness matrix K, theflexibility matrix F and the dynamic characteristics ofthe structure.

K�Mf�fTM�M��n

i�1

w2ififT

i �M (11)

F�f�−1fT��n

i�1

1w2

i

fifTi (12)

where wi is the ith modal frequency, �=diag(1/w2) is the


Fig. 5. MAC factors in function of static loading steps: symmetricalloading configuration.

modal stiffness matrix, fi is the ith mode shape, and nis the number of measured modes.

If Fu and Fd are the flexibility matrices correspondingto the undamaged and damaged states of the structure,a damage indicator matrix �F can be defined as the dif-ference between the first two matrices:

�F�Fu�Fd (13)

Each column of the �F matrix corresponds to themeasurement locations on the structure. The damagedetection and localisation is made according to the detec-tion and locatioln is made accordingly to the maximumabsolute values of each column in the �F matrix. Thecolumn having the higher absolute difference thus corre-sponds to the degree of freedom affected by the damage.

3. Experimental work

3.1. Measurement set-up

The experimental tests intend to evaluate the corre-lation between the progressive cracking process in RCbeams and the resulting changes that can be observed in

Fig. 6. MAC factors in function of static loading steps: asymmetricalloading configuration.

the DSCs. Two types of experiments are combined: thestatic loading test and the dynamic measurements, thefirst to gradually introduce the crack damage in the RCbeams and the second to determine the dynamic charac-teristics.

The test structures are RC beams of 6 m length havinga constant rectangular cross-section (250×200 mm). Thebeams are reinforced with six longitudinal steel (S500)bars of 16 mm diameter, equally distributed over tensionand compression side and transverse reinforcement con-sisting of stirrups with 8 mm diameter placed at each200 mm along the beams.

The crack damage is induced in different steps corre-sponding to an increase of the loading amplitude. Thethree-point and four-point bending test configurationsare adopted. The loading configuration is either sym-metric (first beam) or asymmetric (second beam). Fig. 1shows the two set-up configurations. The two loadingconfigurations are different and allow us to introduce asymmetrical or an asymmetrical (case a or b) test, withthe beams simply supported. The positions of the sup-ports are calculated to minimise the bending moment dueto the weight of the beams and in this manner to avoidcrack appearance before the application of load. At eachintermediate load step, force (load) and vertical displace-


ments are measured and visual inspection is performedto detect cracks. Tables 1 and 2 give a survey of thestatic loading steps.

The dynamic measurements are performed to deter-mine the dynamic characteristics of the beams. Aftereach static loading step, the beams are unloaded and sup-ports are removed. The beams are then suspended usingfour elastic springs attached at the theoretical nodalpoints of the first bending mode during all the dynamicmeasurements. The springs are used to simulate the free–free boundary conditions. Sixty-two measurement pointsare chosen on the top surface of the beams as shown inFig. 2. These points are divided into two rows of 31points on each side of the top surface in order to detectboth torsion and bending vibration modes. The vibrationforces are generated using an electromagnetic shaker(MB MODAL 50 A). The vibration responses are cap-tured using 12 accelerometers (PCB 338A35 and

Fig. 7. 1-COMAC in function of measurement points: symmetrical loading configuration.

338B35) in six series of 10 simultaneous measurements.Two accelerometers are conserved fixed as referencemeasurements on points 1 and 62. The force and acceler-ation signals measured are sent through a tape. The tapeis afterward replayed in the laboratory reproducing ana-logue signals that can be acquired and treated by themodal analysis system CADA-X (Leuven MeasurementsSystem). Only vertical eigenmodes are investigated.

The sampling frequency is chosen to be high enoughto avoid the problem of aliasing for the analysed fre-quency ranges. The use of a self-windowing pseudo-ran-dom excitation signal allows the problem of leakage tobe overcome.

3.2. Results and discussions

3.2.1. Eigenfrequency evolutionFigs. 3 and 4 show the relative variations of the eigen-

frequencies in function of static loading steps. Each load-


Fig. 8. 1-COMAC in function of measurement points: asymmetrical loading configuration: case (a).

ing step corresponds to the level of the crack damageintroduced in the beams. Fig. 3 corresponds to the sym-metrical crack pattern and Fig. 4 corresponds to theasymmetrical case.

It appears from these figures that all the investigatedeigenfrequencies are affected by the accumulation ofcracks in the same manner for both the considered staticloading configurations. The eigenfrequencies decrease.This evolution reveals the decrease of the stiffness in thedamaged sections of the beams. This is ob- ?? flexuralrigidity EI decreases and the resonant frequencies beingrelated to rigidity vary in the same way. It should alsobe noted that the decrease of eigenfrequencies remainsmonotonical during the cracking process in spite of thedifference between the set-up configurations (location ofthe cracked sections: cf. Tables 1 and 2 and Fig. 1).

For the symmetrical cracking configuration (beam 1),the first bending mode is the most affected by the dam-

age. The relative drop in frequency reaches 6% after theappearance of the first crack. This can be explained bythe fact that the cracked zone is located on the part ofthe beam where this mode has the higher vibrationamplitude.

In the asymmetrical cracking configuration (beam 2),the second bending mode is the most affected. In thiscase, the cracked zone is located around one of the twonodal lines of the first eigenmode but in the zone wherethe vibration amplitude of the second eigenmode ishigher. A frequency drop of about 3% is observed afterthe appearance of the first crack for the second eigen-mode. The difference in the shift value and in the mostaffected eigenmode denotes the difference in the crackdamage location.

From these results, one may say that the evolution ofeigenfrequencies is not affected by cracking location inthe RC beams. The beams react in the same way wher-


Fig. 9. 1-COMAC in function of measurement points: asymmetrical loading configuration: case (b).

ever the damaged section appears in the beams. The eig-enmodes are not affected in the same way for the twocases, but nevertheless the damage location remains dif-ficult to predict using the eigenfrequency evolution only.This observation makes this modal parameter a suitableone but complementary methods are needed to locate thedamaged areas.

3.2.2. MAC evolutionsFigs. 5 and 6 show the evolutions of the MAC factors

in function of the damage level introduced in the RCbeams (static loading steps). The last static loading stepcorresponds to the failure of the beams.

In contrast to the results on eigenfrequencies, the evol-utions of the MAC factors are not monotonical. More-over, the observed evolutions are different for the twotested beams:

3.2.2.1. Symmetrical crack configuration In this case,a decreasing tendency is observed in the evolution of

the MAC factors, for each considered eigenmode. Thisdecrease expresses the alteration of the RC beams by thedecrease of the rigidity in certain zones of the structure.However, the decrease is not monotonical and makes theinterpretation of the obtained results more difficult. Thenon-monotonical evolution can be explained by the highsensitivity of the MAC factors to measurement errors.These factors being calculated from the local displace-ment amplitudes of the mode shapes, a measurementerror occurring in one point results at once in a dip of theMAC factors, even though the displacement amplitudesremain the same in the other points of the structure,which leads to a wrong interpretation as to the state ofhealth of the structure. The user has thus to be carefulwhen processing these parameters. The good quality ofmode shape measurements is necessary to obtain accur-ate values of MAC factors.

3.2.2.2. Asymmetrical crack configuration In thiscase, the evolution of the MAC factors is surprising. The


Fig. 10. Damage index in function of measurement points: symmetrical loading configuration.

MAC factors decrease for the first set-up configuration(case (a) of the asymmetrical configuration), and thedecrease is nearly monotonical for each considered eig-enmode expressing the alteration in rigidity of the struc-ture. Case (b) results in contrast in the increase of theMAC factors as if the structure has been restored. Thislast observation was not expected.

According to this result, one may conclude that theasymmetrical crack damage occurring in the RC beamcauses a decrease of the MAC factors expressing theasymmetrical alteration of the structure. But, by makingthe crack damage symmetrical, the MAC factors increaseagain. Nevertheless, it should be noted that the MACfactors never reach the reference values (undamagedstate) again. The remaining difference expresses the glo-bal alteration in rigidity of the RC beam.

From these observations, one can say that symmetrical

damage occurring in a structure has less influence on theMAC factors evolution than the asymmetrical one.

3.2.3. COMAC evolutionFigs. 7–9 show the evolutions of the COMAC factors

for the two tested RC beams. Fig. 7 corresponds to thesymmetrically damaged beam, Fig. 8 corresponds to thecase (a) of the asymmetrically damaged beam and Fig.9 corresponds to the case (b) of the asymmetrically dam-aged beam. The graphs presented show the (1-COMAC)evolution for reason of visibility. All the figures presentthe evolution of the (1-COMAC) values in differentmeasurement points: the graphs on each figure expressesthe evolution of (1-COMAC) values for different staticloading step, compared to the reference state of the struc-ture.

Eigenfrequencies and MAC factors can only indicate


Fig. 11. Damage index in function of measurement points: asymmetrical loading configuration: case (a).

whether damage exists but, in practice, it is important toidentify where the damage is localized. To resolve thisproblem, the use of mode shape derivatives providinginformation for individual measurement points isadequate. The COMAC factor is a candidate for this kindof parameter. The observations drawn from the obtainedresults may be divided into two parts.

3.2.3.1. Symmetrical crack configuration Fig. 7shows the evolution of the (1-COMAC) value in the caseof the symmetrical static loading configuration. Amaximum drop of 15% is observed in the COMAC fac-tor which corresponds to the higher statistic loadingamplitude (load step 6). One can point out that the maxi-mal drops are localised in the points 11 and 21 wherethe static loading is applied. This is striking for all theconsidered loading steps permitting us to identify the

damaged sections. Nevertheless, it is difficult to see thatthe whole zone running from point 11 to 21 is damagedin the same way, because the COMAC factor is notaffected in the same way in all points between 11 and21. Looking at all the graphs, the drop of the COMACruns from 0.8% to 15% at the failure of the beam(step 6).

3.2.3.2. Asymmetrical crack configuration Fig. 8shows the evolution of the (1-COMAC) value for thecase (a) of an asymmetrical static loading configuration.Three static loading steps are presented: (st1a), (st2a)and (st5a). The maximum drop of the COMAC factor isof the order of 4.5% which is obtained after the finalstatic loading step (st5a). This drop is situated at theloading point 11 indicating the damaged section of thebeam. On the other hand, looking at Fig. 9 corresponding


Fig. 12. Damage index in function of measurement points: asymmetrical loading configuration: case (b).


Fig. 13. Flexibility change in function of measurement points: symmetrical loading configuration.

to the case (b), it is difficult to detect the new damagedsection. The drop in the COMAC remains located in theneighbourhood of the point 11. So, no indication is givenconcerning the second damaged section correspondingto the loading point 21 (see Fig. 2).

From this observation, one can say that it is difficultto detect two different damaged sections in the samebeam with different severity of damage.

3.2.4. Strain energy evolutionFigs. 10–12 show the evolutions of the damage indi-

ces for the two tested RC beams. Two cases are also con-sidered.

3.2.4.1. Symmetrical crack configuration Thepresented damage indices correspond to the static load-ing steps (st1), (st3), (st5) and (st6). The damage indicesrun from 9 to 65 (cf. Fig. 10). Although the damage runsfrom point 11 to point 21, this is not reflected in the

evolution of the damage indices. Once again, the valuesof the damage indices do not reflect the severity of thedamage. Furthermore, the section corresponding to point11 appears not to be affected by the damage.

3.2.4.2. Asymmetrical crack configuration Thepresented damage indices on Fig. 11 correspond to thestatic loading steps (st1a), (st2a), (st3a) and (st5a). Thosepresented on Fig. 12 correspond to the static loadingsteps (st1b), (st2b), (st5b) and (st7b). For the case (a),the damage indices vary from 35 to about 60 and allowus to identify clearly the damaged section (point 11).Nevertheless, the evolutions of the damage indices donot allow us to follow the severity of the damage,because the final static loading step shows a value thatis smaller than for the first loading step. The case (b)also permits us to identify the second damaged section(point 21) (cf. Fig. 12). In contrast to the COMAC fac-tors, it is possible to identify the two damaged sections


Fig. 14. Flexibility change in function of measurement points: asymmetrical loading configuration: case (a).

simultaneously. Nevertheless, as for the case (b), theseverity of the damage remains difficult to predict.

From the results described above, the damage indicesare more precise in the identification of the damagedzones in RC beams than the COMAC factors in the caseof local damages. The difficulty remains when the dam-age is spread out over a certain length of the RC beam.

3.2.5. Flexibility methodFigs. 13–15 present the change in flexibility matrices

of the tested RC beams. The general shapes of the graphscorrespond well to the pattern mentioned in Ref. 6 forthe beams tested in free–free boundary conditions.

Fig. 14 shows the change in flexibility matricesobtained from the case (a) of the asymmetrically dam-aged beam. The change in flexibility runs from about400 to about 1400, allowing damage detection. But,nothing permits an indication of where the damage

occurs. The maximum observed changes are not fixed inone location and make it difficult to interpret theobtained results. This may be explained by the fact thatthe damage in RC structures is never localised but spreadover a certain zone.

In the case (b), the change in flexibility runs from 700to about 4500 and shows that the damage is more severethan in the case (a) (cf. Fig. 15). Once again, the localis-ation remains difficult. The damaged zone (point 21) cannot be identified.

For the RC symmetrically damaged beam, the changein flexibility runs from 400 to about 2000 (Fig. 13), andagain it is difficult to locate the damaged zone.

From the above described results, one can concludethat the change in flexibility matrices may permit detec-tion of damage in RC beams but localisation is very dif-ficult. The difficulty is related to the nature of damagein RC beams: the damage is spread over a distance and


Fig. 15. Flexibility change in function of measurement points: asymmetrical loading configuration: case (b).


causes changes in flexibility matrices even in zoneswhere the static loading is not applied.

4. Conclusions

Two RC beams are tested within the framework of thedevelopment of a health monitoring system for damagedetection in civil engineering structures based on thechanges of DSCs. The tests consist in subjecting the RCbeams to progressive cracking processes and in measur-ing the changes observed in the dynamic parameters.Different methods are used for damage detection andlocalisation.

It appears from this analysis that:

� the eigenfrequencies are affected by accumulation ofcracks in the RC beams but their evolutions are notinfluenced by the crack damage locations. It shouldbe noted that the decrease of eigenfrequencies is mon-otonical, that allows the severity of the damage tobe followed;

� the MAC factors are, in contrast, less sensitive tocrack damage compared with eigenfrequencies butgive an indication of the symmetrical or asymmetricalnature of the damage.

� With the COMAC factor evolution, it is possible todetect and locate damage in the tested RC beams. Oneshould notice nevertheless the difficulty to follow thedamage severity and spreading, which are notreflected in the drop of the COMAC factors.

� The change in flexibility matrices allows also detec-tion of the crack damage in RC beams, but the dam-age localisation is difficult. The difficulty comes fromthe nature of the crack damage in RC beams: thecracks are not only limited in the section where thestatic load is applied but is spread over a certain dis-tance on both sides of the loaded section, this causesalso the changes in flexibility matrices in the sectionswhere the static load is not applied, which makes thedamage localisation very difficult.

� The damage indices also allow the damage to bedetected and located. They appear to be more precisein the identification of the cracked zones in RC beamsthan the COMAC factors and the flexibility matricesin the case of local damage. As for the other methods,the difficulty remains when the damage is spread outover a certain length of the RC beam, it becomes inthis case very difficult to identify the damaged zones.

From this study, all the above-mentioned methods allowthe detection of crack damage in RC beams. The damageseverity can be followed using the eigenfrequency evol-utions. Damage localisation is possible for localiseddamage and the strain energy method appears to be moreprecise than the others.

Acknowledgements

This work is a part of a research project G-0243.96sponsored by the Flanders Fund for Scientific Researchof Belgium. Its financial support is gratefully acknowl-edged.

References

[1] Ndambi JM, Peeters B, Maeck J, De Visscher J, Wahab MA, DeRoeck G, De Wilde WP. Comparison of techniques for modalanalysis of concrete structures. Eng Struct 2000;22(9):1159–66.

[2] Doebling SW, Farrar CR, Prime MB, Shevitz DW. Damageidentification and health monitoring of structural and mechanicalsystems from changes in their vibration characteristics. ‘A litera-ture review’ . New Mexico: Los Alamos National Laboratory,1996.

[3] Zakic BD. Vibrations in diagnosis of damages in concretebridges. In: Proceedings of the Second RILEM International Con-ference on Diagnosis of Concrete Structures, Strbske pleso, Slo-vakia, 1996:320–3.

[4] Identification approach based on element modal strain energy. In:Proceedings of the International Conference on Noise andVibration Engineering (ISMA23). Leuven, Belgium, 1998:107–114.

[5] Cornwell P, Doebling S, Farrar CR. Application of the strainenergy damage detection method to plate-like structures. J SoundVib, 224(2):359–74.

[6] Pandey AK, Biswas M, Samman MM. Damage detection fromchanges in curvature mode shapes. J Sound Vib1991;145(2):321–32.

[7] CADA-X, Modal Analysis, User manual, Rev. 3.4. Lms Inter-national, 1996.

[8] Heylen W, Lammens S, Sas P. Modal analysis theory and testing.KULeuven: Department of Mechanical Engineering, 1995.

[9] Alampalli S, Fu G, Evertt WD. Measuring bridge vibration fordetection of structural damage. Department of Transportation,Research Project 165, New York, 1995.

[10] Pirner M, Pospisil S. Influence of damage on natural vibrationmodes of R.C. plates. In: Proceedings of the Second RILEMInternational Conference on Diagnosis of Concrete Structures,Strbske pleso, Slovakia, 1996:238–43.

[11] Pandey AK, Biswas M. Damage detection in structures usingchanges in flexibility. J Sound Vib 1994;169(1):3–17.

comac method damage detection

Documents