Damage assessment in reinforced concrete beams using eigenfrequencies and mode shape derivatives

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

Damage assessment in reinforced concrete beams using eigenfrequencies and mode shape derivatives J.-M. Ndambi a b

a,*

, J. Vantomme a, K. Harri

b

Royal Military Academy, Civil Engineering Department, Av. de la Renaissance 30, B-1000 Brussels, Belgium 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 civil engineering structures, based on the changes of dynamic characteristics. As a part of this research, reinforced concrete beams of 6 meters length are subjected to progressing cracking introduced in different steps. The damaged sections are located in symmetrical or asymmetrical positions according to the beam tested. The damage assessment consists in relating the changes observed in the dynamic 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 evolutions are not influenced by the crack damage locations; they decrease with the crack damage accumulation. The MAC factors are less sensitive to crack damage compared to eigenfrequencies, but give an indication of the symmetrical or asymmetrical nature of the induced crack damage. Next to this, the COMAC factors, the strain energy evolution and the changes in flexibility matrices are also examined as to their capability for detection and location of damage in the RC beams, the strain energy method appears to be 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; Flexibility matrix

1. Introduction Nowadays, modal testing methods are widely used as non-destructive tools in many engineering applications to detect and evaluate damage, including civil engineering applications. This is due to the development of new powerful systems in data acquisition and signal processing, allowing to determine accurately the dynamic system characteristics (DSCs). Nevertheless, the user has to be careful because some non-controllable problems related to measurement quality and accuracy may lead to wrong interpretation of the obtained measurements. Some of these problems are mentioned in Ref. 1. However, the problem remains to establish a correct correlation 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).

the damage appearance, the detectable severity and the location. The basic idea of damage evaluation techniques based on vibrations is that the dynamic characteristics are functions of the physical properties of the structure and therefore any change in these properties caused by damage results in the change of DSCs [2]. In concrete structures, it has already been proved that damage, either local or global, is associated with structural modifications, which can be observed through changes in dynamic characteristics: eigenfrequencies, modal damping ratios, mode shapes and derivatives [3,4]. This observation makes the damage detection techniques based on changes of DSCs promising in civil engineering applications. The strain energy method was applied by Cornwell et al. [5] to an aluminium plate to detect and locate damage in the plate using a saw to obtain localised damage. They successfully detected and located damage in an aluminium plate using the damage index (see Section 2.2). Pandey et al. [6] used the change in the flexibility

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matrices to detect damage in a (US standard) beam W12×16. The damage was simulated by opening bolts from the splice plates. Once again, the damage was well localised and this damage could be detected and localised by the change observed in the flexibility matrices of the structures. This paper presents additional results obtained in the framework of a research program for evaluation of dynamic techniques for damage detection in reinforced concrete (RC) beams. Increasing levels of crack damage are introduced by subjecting the beams to static loads with increasing amplitude. After each step, dynamic measurements are performed to determine the dynamic characteristics. These characteristics are obtained by performing curve fit procedures on a number of measured frequency response functions (FRFs). The Frequency Domain direct Parameter Identification (FDPI) technique implemented on the CADA-X [7] system is used for the dynamic characteristics estimation. The universal files produced by the CADA-X system are imported to MATLAB software in order to calculate the strain energy distribution and the flexibility matrices of the beams. The comparison of the set of measurements in damaged and undamaged state of the beams allows the sensitivity analysis of eigenfrequencies, modal assurance criterion (MAC) factors, coordinate modal assurance to crack damage in RC beams. The 1-D strain energy method is applied to the beams to detect and locate the crack damage. For the last method, the damage index is used for the damage detection.

n

MAC(jk)⫽

冘

Next to the basic modal parameters of structures such as eigenfrequencies, modal damping ratios and mode shapes, some derived coefficients obtained from these parameters can also be useful for damage detection in structures. The MAC and COMAC factors may be mentioned in this category. These factors are derived from mode shapes and express the correlation between two measured mode shapes obtained from two sets of tests. The MAC and COMAC factors being linked to mode shapes, sufficient number of degrees of freedom (number of measurement points) are needed to obtain accurate quantities. The next paragraph gives a short theoretical description of the MAC and COMAC factors [8–10]. Let [⌿A] and [⌿B] be the first and second sets of measured mode shapes in matrix form of sizes n×mA and n×mB respectively, with mA and mB the numbers of modes shapes considered in the respective sets and the number of measurement points. The MAC factor is then defined for the modes shapes j and k as follows:

[⌿A]ji [⌿B]ki兩2

i⫽1

n

(1)

冘 n

([⌿A]ji )2

i⫽1

([⌿B]ki)2

i⫽1

where j=1%mA and k=1%mB. [⌿A]ji and [⌿B]ki are the ith components of the modes [⌿A]j and [⌿B]k , respectively. MAC(j,k) factor indicates the degree of correlation between the jth mode of the first set A and the kth mode of the second set of mode shapes B. The MAC values vary from 0 to 1, with 0 for no correlation and 1 for full correlation. Therefore, if the eigenvectors [⌿A] and [⌿B] are identical, the corresponding MAC value will be close to 1, thus indicating the full correlation between the two modes. The deviation of these factors from 1 could be interpreted as a damage indicator in structures. The COMAC factors are generally used to identify where the mode shapes of a structure from two sets of measurements do not correlate. If the modal displacement in a coordinate i from two sets of measurements are identical, the COMAC factor is close to 1 for this coordinate. A large deviation from unity can be interpreted as damage indication in the structure. For the coordinate i of a structure and using m mode shapes, the COMAC factor is defined as follows:

冘 m

[ COMAC(i)⫽ m

冘

2. Damage detection: theoretical aspects 2.1. Modal Assurance Criterion (MAC) and Coordinate Modal Assurance Criterion (COMAC)

冘

兩

兩[⌿A]ji [⌿B]ji 兩]2

j⫽1

冘

(2)

m

j 2 A i

([⌿ ] )

j⫽1

j 2 B i

([⌿ ] )

i⫽1

2.2. Strain energy method The starting point of this method is the formulation of the strain energy of a Bernoulli–Euler beam given by the relation (3). Two versions of this method exist: the one applicable to the beam-like structures (1-D) and the other one applicable to the plate-like structures (2-D). The next paragraph resumes some essential elements of the first method [5]. U⫽

冕 冉 冊

∂2w 1 l EI 2 0 ∂x2

2

dx

(3)

where U is the strain energy, EI is the flexural rigidity of the beam, l is the length of the beam and w is the shape function of the beam. If a mode shape yi(x) is considered, the associated strain energy Ui is given by:

冕 冉 冊

1 l ∂2yi Ui⫽ EI 2 0 ∂x2

2

dx

(4)

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Uij ⫽

1 2

冕

aj+1

(EI)j

aj

冉 冊 ∂2yi ∂x2

503

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

冘 N

Fij ⫽1

(7)

j⫽1

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

Considering the damaged structure, the quantities mentioned in Eqs. (4)–(7) can also be quantities obtained from the damaged mode shapes yd of the structure. From these quantities and after some mathematical arrangements and making some assumptions, Eq. (8) can be established [5].

冕 冉 冕冉 冕 冉 冕冉

Table 1 Static loading steps: symmetric configuration

ak

l

Load steps

Loads (kN)

ref st1 st2 st3 st4 st5 st6

(EI)k ⫽ (EI)dk

0 4 6 12 18 24 25.3

0

l

0

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

First set-up (a) Load steps

Loads (kN)

Load steps

Loads (kN)

ref st1a st2a st3a st4a st5a

0 7 10 13 19 25

st1b st2b st3b st4b st5b st6b st7b

8 10 13 19 25 35 53

f dik fik

(8)

∂2yi 2 dx ∂x2

冘 冘 m

f dik

bk ⫽

Second set-up (b)

∂2yi 2 dx ∂x2

⫽

with (EI)k and (EI)dk corresponding to the flexural rigidities of the sub-region k in the undamaged and damaged states, respectively. In order to use all the measured modes shapes m in the calculation, the damage index bk for the sub-region k is defined as:

i⫽1

Table 2 Static loading steps: asymmetric configurations

∂2ydi 2 dx ∂x2

ak+1 ak

冊 冊 冊 冊

∂2ydi 2 dx ∂x2

ak+1

(9)

m

fik

i⫽1

The value of the damage index b allows us to evaluate the health state of the structure. bk expresses the degradation of the flexural rigidity in a certain zone of the structure by the change in strain energy stored in the structure when it deforms in its particular mode shape. In Eq. (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

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Fig. 2.

Dynamic measurements: measurement locations.

Fig. 3. Eigenfrequencies in function of static loading steps: symmetrical configuration.

random variable, a normalised damage index zk is defined as follows: bk−b¯ k (10) zk⫽ sk where b¯ k and sk represent the mean and standard deviation of the damage indices, respectively.

Fig. 4. Eigenfrequencies in function of static loading steps: asymmetrical configuration.

son of the flexibility matrices obtained from two sets of experimental mode shapes. The method is applicable only if the mode shapes are mass-normalized to unity (fTMf=1). If this is the case, the measured flexibility matrices can then be estimated from the mass-normalized mode shapes. The following equations give the relationships between the system stiffness matrix K, the flexibility matrix F and the dynamic characteristics of the structure.

2.3. Flexibility method It has already been proved that the presence of cracks in a structure increases its flexibility. So, the experimental changes observed in the flexibility matrix can be interpreted as a damage indication in the structure and may allow one to evaluate and locate damage [11]. The principle of this method is based on the compari-

冉冘 冊 n

K⫽Mf⍀fTM⬵M

w2i fifTi M

(11)

i⫽1

冘 n

F⫽f⍀−1fT⬵

1 T ff w2 i i i⫽1 i

(12)

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

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505

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

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

modal stiffness matrix, fi is the ith mode shape, and n is the number of measured modes. If Fu and Fd are the flexibility matrices corresponding to the undamaged and damaged states of the structure, a damage indicator matrix ⌬F can be defined as the difference between the first two matrices:

the DSCs. Two types of experiments are combined: the static loading test and the dynamic measurements, the first to gradually introduce the crack damage in the RC beams and the second to determine the dynamic characteristics. The test structures are RC beams of 6 m length having a constant rectangular cross-section (250×200 mm). The beams are reinforced with six longitudinal steel (S500) bars of 16 mm diameter, equally distributed over tension and compression side and transverse reinforcement consisting of stirrups with 8 mm diameter placed at each 200 mm along the beams. The crack damage is induced in different steps corresponding to an increase of the loading amplitude. The three-point and four-point bending test configurations are adopted. The loading configuration is either symmetric (first beam) or asymmetric (second beam). Fig. 1 shows the two set-up configurations. The two loading configurations are different and allow us to introduce a symmetrical or an asymmetrical (case a or b) test, with the beams simply supported. The positions of the supports are calculated to minimise the bending moment due to the weight of the beams and in this manner to avoid crack appearance before the application of load. At each intermediate load step, force (load) and vertical displace-

⌬F⫽Fu⫺Fd

(13)

Each column of the ⌬F matrix corresponds to the measurement locations on the structure. The damage detection and localisation is made according to the detection and locatioln is made accordingly to the maximum absolute values of each column in the ⌬F matrix. The column having the higher absolute difference thus corresponds to the degree of freedom affected by the damage.

3. Experimental work 3.1. Measurement set-up The experimental tests intend to evaluate the correlation between the progressive cracking process in RC beams and the resulting changes that can be observed in

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ments are measured and visual inspection is performed to detect cracks. Tables 1 and 2 give a survey of the static loading steps. The dynamic measurements are performed to determine the dynamic characteristics of the beams. After each static loading step, the beams are unloaded and supports are removed. The beams are then suspended using four elastic springs attached at the theoretical nodal points of the first bending mode during all the dynamic measurements. The springs are used to simulate the free– free boundary conditions. Sixty-two measurement points are chosen on the top surface of the beams as shown in Fig. 2. These points are divided into two rows of 31 points on each side of the top surface in order to detect both torsion and bending vibration modes. The vibration forces are generated using an electromagnetic shaker (MB MODAL 50 A). The vibration responses are captured using 12 accelerometers (PCB 338A35 and

Fig. 7.

338B35) in six series of 10 simultaneous measurements. Two accelerometers are conserved fixed as reference measurements on points 1 and 62. The force and acceleration signals measured are sent through a tape. The tape is afterward replayed in the laboratory reproducing analogue signals that can be acquired and treated by the modal analysis system CADA-X (Leuven Measurements System). Only vertical eigenmodes are investigated. The sampling frequency is chosen to be high enough to avoid the problem of aliasing for the analysed frequency ranges. The use of a self-windowing pseudo-random excitation signal allows the problem of leakage to be overcome. 3.2. Results and discussions 3.2.1. Eigenfrequency evolution Figs. 3 and 4 show the relative variations of the eigenfrequencies in function of static loading steps. Each load-

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

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Fig. 8.

507

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

ing step corresponds to the level of the crack damage introduced in the beams. Fig. 3 corresponds to the symmetrical crack pattern and Fig. 4 corresponds to the asymmetrical case. It appears from these figures that all the investigated eigenfrequencies are affected by the accumulation of cracks in the same manner for both the considered static loading configurations. The eigenfrequencies decrease. This evolution reveals the decrease of the stiffness in the damaged sections of the beams. This is ob- ?? flexural rigidity EI decreases and the resonant frequencies being related to rigidity vary in the same way. It should also be noted that the decrease of eigenfrequencies remains monotonical during the cracking process in spite of the difference between the set-up configurations (location of the 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 the appearance of the first crack. This can be explained by the fact that the cracked zone is located on the part of the beam where this mode has the higher vibration amplitude. In the asymmetrical cracking configuration (beam 2), the second bending mode is the most affected. In this case, the cracked zone is located around one of the two nodal lines of the first eigenmode but in the zone where the vibration amplitude of the second eigenmode is higher. A frequency drop of about 3% is observed after the appearance of the first crack for the second eigenmode. The difference in the shift value and in the most affected eigenmode denotes the difference in the crack damage location. From these results, one may say that the evolution of eigenfrequencies is not affected by cracking location in the RC beams. The beams react in the same way wher-

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Fig. 9.

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

ever the damaged section appears in the beams. The eigenmodes are not affected in the same way for the two cases, but nevertheless the damage location remains difficult to predict using the eigenfrequency evolution only. This observation makes this modal parameter a suitable one but complementary methods are needed to locate the damaged areas. 3.2.2. MAC evolutions Figs. 5 and 6 show the evolutions of the MAC factors in function of the damage level introduced in the RC beams (static loading steps). The last static loading step corresponds to the failure of the beams. In contrast to the results on eigenfrequencies, the evolutions of the MAC factors are not monotonical. Moreover, the observed evolutions are different for the two tested beams:

the MAC factors, for each considered eigenmode. This decrease expresses the alteration of the RC beams by the decrease of the rigidity in certain zones of the structure. However, the decrease is not monotonical and makes the interpretation of the obtained results more difficult. The non-monotonical evolution can be explained by the high sensitivity of the MAC factors to measurement errors. These factors being calculated from the local displacement amplitudes of the mode shapes, a measurement error occurring in one point results at once in a dip of the MAC factors, even though the displacement amplitudes remain the same in the other points of the structure, which leads to a wrong interpretation as to the state of health of the structure. The user has thus to be careful when processing these parameters. The good quality of mode shape measurements is necessary to obtain accurate values of MAC factors.

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

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

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Fig. 10.

509

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 the decrease is nearly monotonical for each considered eigenmode expressing the alteration in rigidity of the structure. Case (b) results in contrast in the increase of the MAC factors as if the structure has been restored. This last observation was not expected. According to this result, one may conclude that the asymmetrical crack damage occurring in the RC beam causes a decrease of the MAC factors expressing the asymmetrical alteration of the structure. But, by making the crack damage symmetrical, the MAC factors increase again. Nevertheless, it should be noted that the MAC factors never reach the reference values (undamaged state) again. The remaining difference expresses the global alteration in rigidity of the RC beam. From these observations, one can say that symmetrical

damage occurring in a structure has less influence on the MAC factors evolution than the asymmetrical one. 3.2.3. COMAC evolution Figs. 7–9 show the evolutions of the COMAC factors for the two tested RC beams. Fig. 7 corresponds to the symmetrically damaged beam, Fig. 8 corresponds to the case (a) of the asymmetrically damaged beam and Fig. 9 corresponds to the case (b) of the asymmetrically damaged beam. The graphs presented show the (1-COMAC) evolution for reason of visibility. All the figures present the evolution of the (1-COMAC) values in different measurement points: the graphs on each figure expresses the evolution of (1-COMAC) values for different static loading step, compared to the reference state of the structure. Eigenfrequencies and MAC factors can only indicate

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Fig. 11.

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

whether damage exists but, in practice, it is important to identify where the damage is localized. To resolve this problem, the use of mode shape derivatives providing information for individual measurement points is adequate. The COMAC factor is a candidate for this kind of parameter. The observations drawn from the obtained results may be divided into two parts.

damaged sections. Nevertheless, it is difficult to see that the whole zone running from point 11 to 21 is damaged in the same way, because the COMAC factor is not affected in the same way in all points between 11 and 21. Looking at all the graphs, the drop of the COMAC runs from 0.8% to 15% at the failure of the beam (step 6).

3.2.3.1. Symmetrical crack configuration Fig. 7 shows the evolution of the (1-COMAC) value in the case of the symmetrical static loading configuration. A maximum drop of 15% is observed in the COMAC factor which corresponds to the higher statistic loading amplitude (load step 6). One can point out that the maximal drops are localised in the points 11 and 21 where the static loading is applied. This is striking for all the considered loading steps permitting us to identify the

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

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

Fig. 12.

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

511

512

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Fig. 13.

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

to the case (b), it is difficult to detect the new damaged section. The drop in the COMAC remains located in the neighbourhood of the point 11. So, no indication is given concerning the second damaged section corresponding to the loading point 21 (see Fig. 2). From this observation, one can say that it is difficult to detect two different damaged sections in the same beam with different severity of damage. 3.2.4. Strain energy evolution Figs. 10–12 show the evolutions of the damage indices for the two tested RC beams. Two cases are also considered. 3.2.4.1. Symmetrical crack configuration The presented damage indices correspond to the static loading steps (st1), (st3), (st5) and (st6). The damage indices run from 9 to 65 (cf. Fig. 10). Although the damage runs from point 11 to point 21, this is not reflected in the

evolution of the damage indices. Once again, the values of the damage indices do not reflect the severity of the damage. Furthermore, the section corresponding to point 11 appears not to be affected by the damage. 3.2.4.2. Asymmetrical crack configuration The presented damage indices on Fig. 11 correspond to the static loading steps (st1a), (st2a), (st3a) and (st5a). Those presented on Fig. 12 correspond to the static loading steps (st1b), (st2b), (st5b) and (st7b). For the case (a), the damage indices vary from 35 to about 60 and allow us to identify clearly the damaged section (point 11). Nevertheless, the evolutions of the damage indices do not allow us to follow the severity of the damage, because the final static loading step shows a value that is 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 factors, it is possible to identify the two damaged sections

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Fig. 14.

513

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

simultaneously. Nevertheless, as for the case (b), the severity of the damage remains difficult to predict. From the results described above, the damage indices are more precise in the identification of the damaged zones in RC beams than the COMAC factors in the case of local damages. The difficulty remains when the damage is spread out over a certain length of the RC beam. 3.2.5. Flexibility method Figs. 13–15 present the change in flexibility matrices of the tested RC beams. The general shapes of the graphs correspond well to the pattern mentioned in Ref. 6 for the beams tested in free–free boundary conditions. Fig. 14 shows the change in flexibility matrices obtained from the case (a) of the asymmetrically damaged beam. The change in flexibility runs from about 400 to about 1400, allowing damage detection. But, nothing permits an indication of where the damage

occurs. The maximum observed changes are not fixed in one location and make it difficult to interpret the obtained results. This may be explained by the fact that the damage in RC structures is never localised but spread over a certain zone. In the case (b), the change in flexibility runs from 700 to about 4500 and shows that the damage is more severe than in the case (a) (cf. Fig. 15). Once again, the localisation remains difficult. The damaged zone (point 21) can not be identified. For the RC symmetrically damaged beam, the change in flexibility runs from 400 to about 2000 (Fig. 13), and again it is difficult to locate the damaged zone. From the above described results, one can conclude that the change in flexibility matrices may permit detection of damage in RC beams but localisation is very difficult. The difficulty is related to the nature of damage in RC beams: the damage is spread over a distance and

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Fig. 15.

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

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causes changes in flexibility matrices even in zones where the static loading is not applied. 4. Conclusions Two RC beams are tested within the framework of the development of a health monitoring system for damage detection in civil engineering structures based on the changes of DSCs. The tests consist in subjecting the RC beams to progressive cracking processes and in measuring the changes observed in the dynamic parameters. Different methods are used for damage detection and localisation. It appears from this analysis that: 앫 the eigenfrequencies are affected by accumulation of cracks in the RC beams but their evolutions are not influenced by the crack damage locations. It should be noted that the decrease of eigenfrequencies is monotonical, that allows the severity of the damage to be followed; 앫 the MAC factors are, in contrast, less sensitive to crack damage compared with eigenfrequencies but give an indication of the symmetrical or asymmetrical nature of the damage. 앫 With the COMAC factor evolution, it is possible to detect and locate damage in the tested RC beams. One should notice nevertheless the difficulty to follow the damage severity and spreading, which are not reflected in the drop of the COMAC factors. 앫 The change in flexibility matrices allows also detection of the crack damage in RC beams, but the damage localisation is difficult. The difficulty comes from the nature of the crack damage in RC beams: the cracks are not only limited in the section where the static load is applied but is spread over a certain distance on both sides of the loaded section, this causes also the changes in flexibility matrices in the sections where the static load is not applied, which makes the damage localisation very difficult. 앫 The damage indices also allow the damage to be detected and located. They appear to be more precise in the identification of the cracked zones in RC beams than the COMAC factors and the flexibility matrices in the case of local damage. As for the other methods, the difficulty remains when the damage is spread out over a certain length of the RC beam, it becomes in this case very difficult to identify the damaged zones.

515

From this study, all the above-mentioned methods allow the detection of crack damage in RC beams. The damage severity can be followed using the eigenfrequency evolutions. Damage localisation is possible for localised damage and the strain energy method appears to be more precise than the others.

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

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