Dynamic identification of a reinforced concrete damaged bridge

Mechanical Systems and Signal Processing 25 (2011) 2990–3009

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Dynamic identification of a reinforced concrete damaged bridge Michele Dilena, Antonino Morassi , Marina Perin Dipartimento di Ingegneria Civile e Architettura, Universita degli Studi di Udine, Via Cotonificio 114, 33100 Udine, Italy

a r t i c l e in f o

abstract

Article history: Received 2 April 2011 Accepted 28 May 2011 Available online 1 July 2011

The results of a series of harmonically forced tests carried out on a reinforced concrete single-span bridge subjected to increasing levels of damage are interpreted in this paper. The deck structure of the bridge consists of a slab and three simply supported beams. The damage is represented by a series of notches made on a lateral beam to simulate the effect of incremental concentrated damage. The variation of lower natural frequencies shows an anomalous increase in the transition from one intermediate damage configuration to the next ones. Vibration mode shapes show an appreciable asymmetry in the reference configuration, despite the nominal symmetry of the bridge. A justification of this unexpected dynamic behavior is presented in this paper. The analysis is based on progressive identification of an accurate finite element model of the reference configuration and on reconstruction of damage evolution from natural frequency and vibration mode measurements. Changes in modal curvature of the first two vibration modes evaluated along the main beams are successfully used to identify the location of the damage. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Dynamic tests Bridges Structural identification Finite element models Damage localization Concrete

1. Introduction Damage changes the vibratory behavior of a structure and, therefore, structural diagnostics based on dynamic methods has potentially great importance in engineering applications [3,4,13,14,22]. Recent technological progress has generated extremely accurate and reliable experimental methods, enabling a good estimate of changes in the dynamic behavior of a structural system caused by possible damage. Although experimental techniques are now well-established, the interpretation of measurements still lags somewhat behind. This particularly concerns identification and structural diagnostics by dynamic data due to their nature of inverse problems in vibration [8]. Indeed, in these applications one wishes to determine some mechanical properties of a system on the basis of measurements of its response, in part exchanging the role of the unknowns and data compared to the direct problems of structural analysis. Hence, concerns typical of inverse problems arise, such as non-uniqueness or non-continuous dependence of the solution on the data. When identification techniques are applied to the study of real-world structures, additional obstacles arise given the complexity of structural modeling, the inaccuracy of the analytical models used to interpret experiments, measurement errors and incomplete field data. Furthermore, the results of the theoretical mathematical formulation of problems of identification and diagnostics, given the present state-of-knowledge in the field, focus on quality, while practical needs often require more specific estimates of quantities to be identified. It is probably because of these difficulties that a limited number of studies have investigated so far the effect of damage on modal parameters of full-scale bridges and have developed suitable strategies for damage identification. Without claim of completeness, here we recall the interesting researches developed in [2,7,9–12,18–21,25]. A critical review of the

 Corresponding author. Tel.: þ39 0432 558739; fax: þ 39 0432 558700.

E-mail addresses: [email protected] (M. Dilena), [email protected] (A. Morassi), [email protected] (M. Perin). 0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.05.016

M. Dilena et al. / Mechanical Systems and Signal Processing 25 (2011) 2990–3009

Nomenclature Ec fd ferr K L

Young’s modulus of the concrete spline function objective function to be minimized elastic stiffness span length

pr Rck uðrÞ k

g

d du , dd

2991

undamped circular frequency for mode r cubic strength of concrete kth component of the rth mode shape volume mass density decomposition of the interval [0,L] multipliers

literature shows that there is still no general consensus among experts on the type of data to be taken as good indicator of damage and also on the effectiveness of a diagnostic method rather than another, see, for example, the Introduction of the paper [5]. The main objective of this paper is to give an interpretation of the results of a campaign of dynamic tests carried out on a reinforced concrete bridge under increasing levels of damage. The bridge deck is formed by a slab supported by three longitudinal beams. Harmonically forced dynamic tests were carried out in the reference configuration and in several damage configurations. The damage states were obtained by cutting the downstream beam by means of a hydraulic saw. Some important indications emerged from experiments. First, the trend of natural frequencies to the progress of the damage shows an unexpected increase in the transition from one intermediate damage configuration to the next ones. Second, despite the bridge is nominally symmetric, from vibration mode measurements it was possible to detect a structural asymmetry – mainly in transverse direction – of the reference configuration of the bridge. In the first part of the paper we provide a justification of these two circumstances. The analysis is based on progressive structural identification of reference and damaged configurations from measurements of natural frequencies and vibration modes of the bridge. The second part of the work is addressed to damage localization from mode shape data. In particular, changes in modal curvature of the first two modes evaluated along the main beams were successfully used to identify the location of the damage.

2. Description of the bridge Dogna Bridge is a four-span one-lane concrete bridge. The length of each span is 16.00 m and the lane is 4.00 m width. Fig. 1 shows the span tested during the experiments and considered in the present research. This span is denoted as Dogna Bridge in what follows. The bridge deck is constituted by a reinforced concrete (RC) slab supported by three longitudinal RC beams. The beams are simply supported at the ends on thin metallic sheets and are connected at the supports, at midspan and at span quarters with transverse RC diaphragms. Pier and abutment consist of RC walls and are founded on castin-place concrete piles. Construction of the bridge was completed in 1978. The bridge suffered of an exceptional flood of the Fella River on August 31, 2003. At that time, due to the material deposited upstream, the deck structures of the bridge were involved by the flow of the water, see Fig. 2. A visual inspection conducted on the tested span revealed no apparent deterioration on slab and beams, whereas a state of degradation was noticed on support bearing side pier. For reasons of traffic safety, Dogna Bridge was demolished on May 2008 and has been replaced by a new bridge built about 200 m downstream.

3. Damage scenarios and experimental results The experimental campaign was carried out from April 2 to April 11, 2008. The tested span was made independent of the adjacent span by removing the deck-joint in correspondence of the pier. Moreover, the asphalt overlay of about 0.1 m thickness was also removed before testing. Harmonically forced tests were carried out on the bridge deck in its present condition (reference configuration, indicated by R in the following) and in six damage configurations D1–D6 obtained by cutting the downstream beam as shown in Fig. 1(c). The sequence of notches was produced by means of a hydraulic saw fitted with a diamond disc. The experimental layout is shown in Fig. 1(a). An electric vibrodyne was mounted in vertical direction at one fourth of the upstream beam, near the abutment side. Seventeen piezoelectric accelerometers with vertical axis and one horizontal accelerometer were simultaneously used to determine the deck’s response to the excitation. Deck’s inertance was measured by means of zoom analysis within narrow neighborhoods of the expected natural frequencies values, see [5] for more details. The frequency resolution ranged from 0.02 for lower modes (up to 15 Hz) to 0.04 Hz for higher modes (15–50 Hz). During the experiments a time harmonic force with maximum amplitude of 15 kN has been used. The procedure has been applied for the characterization of the reference and all the damaged configurations D1–D6. Dynamic tests were carried out under similar environmental and weather conditions, so that the influence of temperature and humidity on dynamic modal parameters can be considered negligible.

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Downstream Horizontal accelerometer

A3

A10

A12

A15

A17

A2

A4

A8

A5

A9

3.85

A11

A13

A16

A14

4.01

4.01

Pier

A6 Abutment

1.51

1.42

A7 A1

1.42

1.51

5.85

Damages D1-D6

A18

3.85

16.10 Upstream Accelerometer

Shaker

1.20

0.45 0.98

1.43

0.90 0.15

4.05 0.18

0.90

0.60

0.64 0.35

D1, D2

1.16

0.35

1.16

0.35 0.64

D3, D4

0.60

D5, D6

s ≈ 0.01 0.45

0.45 0.45

Fig. 1. Dogna Bridge: (a) plan view, with indication of the instrumentation; (b) vertical transversal section; (c) damage configurations D1–D6. Lengths in meters.

Modal parameters were extracted from frequency response function measurements. A complete account of the experiments is presented in Ref. [5], which we refer to for more details. In summary, experiments show that natural frequencies are not monotonically decreasing functions of damage severity, see Fig. 3. In particular, from the reference configuration up to damage D3, a decrease of frequency values for increasing degradation is observed. There are two exceptions, namely the second and the fifth natural frequencies increase of 0.03 and 0.30 Hz, respectively, from R to D1 configuration. After that, frequency-damage curves show a tendency to positive increases, approximately from configuration

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2993

Fig. 2. Exceptional flood on August 31, 2003.

9.0

8.0

R D1 D2 D3 D4 D5 D6

26.5

f [Hz]

14.0

f [Hz]

f [Hz]

10.0

13.0

12.0

R D1 D2 D3 D4 D5 D6

35.0

34.0

24.5

R D1 D2 D3 D4 D5 D6

47.5

f [Hz]

f [Hz]

36.0

25.5

R D1 D2 D3 D4 D5 D6

46.5

45.5

R D1 D2 D3 D4 D5 D6

Fig. 3. Experimental natural frequencies vs damage: (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4; (e) Mode 5.

D3 to D5. The trend is quite evident for the first two vibration modes. There is even an abrupt increase in frequency for the fourth mode in the transition from D3 to D4, while the frequency of Mode 3 is increasing with the damage level up to configuration D6. Finally, in the last stage, and with the exception of Mode 3, the values of natural frequencies decrease more or less regularly from configuration D5 up to the most severe damage level D6. This behavior is in contradiction with the general property of natural frequencies for linearly elastic, no dissipative vibrating systems, which must decrease with the increase of structural damage. A justification of this behavior is presented in Section 5.2. The reconstruction of mode shapes revealed a significant peculiarity of the reference configuration of the bridge, see Fig. 4. There is an appreciable loss of symmetry on the mode shapes with respect to the transverse direction of the bridge deck. Amplitudes of oscillation for Mode 1 evaluated on the upstream lateral beam are about 0.80–0.90 the corresponding values measured on the downstream lateral beam. Differences are evident also in higher modes, especially for Modes 3 and 4, see [5] (Figs. 12 and 13). The justification of the lack of symmetry in a nominally symmetric structure has been one of the key issues of the present analysis, see Section 4.4. Experimental results confirm that mode shapes are sensitive to damage. Variations are appreciable in the configuration D1 and become clearly measurable in subsequent damage configurations. Concerning the fundamental bending mode, Mode 1, the spatial shape loses the symmetry in the longitudinal direction and shows a significant increase of the modal components within the damaged region (about 22% for D2–D6, compared to the reference configuration). At the same time, an appreciable decrease of the modal components on the central and upstream beam occurs. The asymmetry in the transversal direction, which was observed in the reference configuration, still remains in the damaged configurations. Changes in Mode 2 are appreciable from configuration D2 and reflect into an accentuation of the amplitudes within the damaged region, with simultaneous reduction in the abutment-side half span. We refer to [5] for a complete account of the results.

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0.20

0.24

0.00

0.00

-0.20 0

L/4

L/2

3L/4

L

-0.24 0

0.20

0.24

0.00

0.00

-0.20 0 0.20

L/4

L/2

3L/4

L

0.00

-0.24 0 0.24

L/4

L/2

3L/4

L

L/4

L/2

3L/4

L

L/4

L/2

3L/4

L

0.00

-0.20 0

L/4 Reference

L/2

3L/4

L

Damage D1

-0.24 0

Damage D3

Damage D5

Fig. 4. Evolution of the first (column (a)) and second (column (b)) normalized vibration mode with respect to damage D1, D3 and D5, for upstream beam (top row), central beam (central row) and downstream beam (bottom row). Abutment: x ¼0; pier: x ¼L.

4. Structural identification of the reference configuration 4.1. FE model updating strategy Structural features of the Dogna Bridge suggest to develop structural identification of the reference configuration on two main steps. The first part of the analysis is devoted to the determination of a suitable class of dynamical FE models of the bridge. The approach is developed by comparison with a refined three-dimensional (3D) solid finite element model (FEM) of the bridge. Next, experimental modal analysis results are used to improve the modeling. Crucial issues are the description of the boundary conditions and the estimate of the cracking effects on the dynamic behavior of the bridge. 4.2. A class of FE models of the deck The two-dimensional (2D) nominal FE model of the deck, M2D, shown in Fig. 5(a), uses a class of discrete Kirchhoff four nodes shell elements with six degrees of freedom at each node. FEs belong to the middle surface of each structural member (slab, longitudinal and transverse beams). Typical size of the FE mesh is about 0.30 m, with aspect ratio between 1 and 1.5. Under the assumption of infinitesimal vibration, the concrete has been assumed to have an isotropic linearly elastic constitutive equation, with Young’s modulus Ec and Poisson’s coefficient nc ¼ 0:20. The average Young’s modulus of the concrete was initially estimated by testing three cores taken from various locations of the slab and beams. Based on these tests, it was assumed Ec ¼32 GPa. The volume mass density g is 2500 kg/m3. In order to keep the uncertainties of structural modeling at the minimum level, Model M2D has been compared with a refined 3D FE model of the bridge deck, Model M3D, under the assumption of free-free boundary conditions. All the longitudinal and transverse beams and the slab of Model M3D are described by solid FEs, see Fig. 5(b). Table 1 collects the lower natural frequencies of both models M2D and M3D. A visual comparison was sufficient to establish the correspondence between mode shapes. With the exception of Mode 5, which corresponds to a plate-like mode of the deck, the discrepancies are almost negligible. It should be noticed that special attention has been devoted to correctly represent inertia properties, avoiding, for example, the multiple evaluation of the mass occurring in overlapping

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2995

Fig. 5. Finite element models of the deck: (a) 2D Model M2D; (b) solid 3D Model M3D.

Table 1 Natural frequencies pn of the bridge deck under free–free boundary conditions estimated by the 2D Model M2D and 3D Model M3D. Values in Hz. Rigid 2D 3D modes are omitted. Dn ¼ 100  ðp3D n pn Þ=pn . Mode order n

Mode shape

Model M2D pn (Hz)

Model M3D pn (Hz)

Error Dn (%)

1 2 3 4 5 6

Torsional Flexural Torsional Flexural Plate-like Torsional

9.374 17.843 24.044 43.121 43.129 45.574

9.479 17.852 24.392 43.483 44.694 46.232

1.1 0.1 1.4 0.8 3.5 1.4

FEs between slab deck and transversal or longitudinal beams, see [17] for details. The total mass of the 2D FE model of the deck is 95,800 kg, whereas the corresponding value in the solid 3D model is 93,500 kg. The small difference (around 3%) is partially responsible of the slight underestimate of the lower natural frequencies obtained by Model M2D. Modeling of the boundary conditions between deck and pier/abutment is a delicate point of the analysis. The flexibility of the abutment and pier supports has been neglected in this analysis. Actually, both the pier and abutment foundations were retrofitted in the past years by enlarging the base foundation and inserting along the contour a series of concrete micropiles. In addition, the experimental results obtained in [5] show that vertical components of mode shapes evaluated at the pier and abutment supports are negligible with respect to the average vertical displacements of the first three vibration modes. The point supports of the main beams on the abutment side were fixed. These constraints were introduced in FE Models M2D and M3D on the lower face of the beams, as it is shown in Fig. 6. Constraints of the beams on the pier side were modeled as supports allowing movement in the longitudinal direction (¼traffic direction) only. As before, these constraints

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Longitudinal view

Transversal view

Plan view

Longitudinal view

Longitudinal view

Transversal view

Plan view

Transversal view

Plan view

Longitudinal view

Transversal view

Plan view

Fig. 6. Longitudinal beam end constraint: example of the central beam. (a) Side abutment in 2D FE model (Model 1); (b) side pier in 2D FE model (Model 1); (c) side abutment in 3D FE model; (d) side pier in 3D FE model.

were placed on the lower face of the beams. The constrained 2D model of the bridge deck will be named Model 1 in the sequel. The eight lower vibration modes obtained by Model 1 are shown in Fig. 7. All the modes, with the exception of the third, have dominant vertical components. Table 2 compares the natural frequencies of Model 1 and of its 3D analog. The agreement is particularly good, with differences less than 1% in the first six modes. Overall, it is believed that the class of 2D FE models is able to adequately represent the dynamic behavior of Dogna Bridge. Within this class of structural models the model updating based on dynamic data will be developed. 4.3. Comparison between Model 1 and experimental data A visual comparison shows that analytical Modes (FEA) 1, 2, 4, 5 and 7 of Model 1 correspond to experimental Modes (EMA) 1, 2, 3, 4 and 5, respectively. Mode 3-FEA involves mainly horizontal vibrations of the bridge deck. Therefore, the lack of correspondence with EMA modes can be explained by noticing that Mode 3 was weakly excited during testing. The lack of correspondence for Mode 6-FEA is probably due to a coincidence between the shaker position and a point of the nodal curve of the vibration mode. Table 3 shows a comparison between the natural frequencies of Model 1 and experimental values. Percentage differences are around 1.5% for EMA Modes 1, 2, 4 and 6–8% for EMA Modes 3 and 5. The comparison between mode shapes shows a fair agreement for the first two modes, see Fig. 8. In spite of the high MAC value for Modes 1 and 2, the loss of symmetry in the transversal direction measured in dynamic testing is not reproduced by Model 1. Differences of the mode shape are significant for EMA Modes 3 and 4, see Fig. 9. Mode 5-EMA is quite correctly reproduced.

M. Dilena et al. / Mechanical Systems and Signal Processing 25 (2011) 2990–3009

Mode 1 (9.33 Hz)

Pie

m rea

st

wn

Do

utm en

Ab

m

ea

str

Up

utm en

utm

t

Mode 5 (35.22 Hz) wn

utm en

Pie

m rea

wn

m

m

ea

str

Up

utm en

Ab

utm

Pie

Mode 8 (53.23 Hz)

m rea

Up

utm en

Pie

m

ea

t

Up

utm en

m

ea

str

Ab

r

m rea

st

wn

Do str

Ab

m

en

r

st

wn

Up t

Mode 7 (44.30 Hz)

Do

r

ea

str

t

t

Pie

m rea

st

Do

ea

Ab

Mode 6 (41.59 Hz) r

st

Do str

m

en

m rea

Up

Up

Ab

r

st

wn

Do

Pie

r

ea

str

t

Mode 4 (28.45 Hz)

Ab

wn

m

Pie

m rea

st

Do

ea

t

Mode 3 (26.50 Hz) r

m rea

wn

Up

Pie

st

Do str

Ab

Mode 2 (13.71 Hz) r

2997

t

Fig. 7. Reference configuration: first eight lower vibration modes of Model 1.

Table 2 Natural frequencies pn of Dogna’s Bridge estimated by the 2D Model (Model 1) and by the corresponding 3D Model. Values in Hz. 2D 3D Dn ¼ 100  ðp3D n pn Þ=pn . Mode order n

Mode shape

Model M2D pn (Hz)

3D model pn (Hz)

Error Dn (%)

1 2 3 4 5 6 7 8

Flexural Torsional Horizontal Flexural Torsional Flexural Plate-like Plate-like

9.334 13.712 26.500 28.448 35.216 41.594 44.301 53.233

9.309 13.710 26.670 28.552 35.355 41.731 45.616 54.334

 0.3 0.0 0.6 0.4 0.4 0.3 2.9 2.0

Table 3 Reference configuration: comparison between experimental (EMA) and analytical (FEA, Models 1 and Model 2) values of natural frequencies pn (in Hz) EMA and MAC values. Dn ¼ 100  ðpFEA Þ=pEMA . n pn n EMA order

1 2 3 4 5

(Hz) pEMA n

9.430 13.487 26.218 34.764 47.171

Model 1

Model 2

FEA order

(Hz) pFEA n

Dn (%)

MAC

FEA order

(Hz) pFEA n

Dn (%)

MAC

1 2 4 5 7

9.334 13.712 28.448 35.216 44.301

 1.0 1.7 8.5 1.3  6.1

99.2 97.2 85.1 76.9 87.3

1 2 4 5 7

8.964 13.522 28.106 34.608 44.030

 4.9 0.3 7.2  0.4  6.7

99.2 97.2 85.3 76.8 87.7

Although Model 1 of the reference configuration of the bridge can be considered satisfactory for most practical engineering applications, an attempt to improve its accuracy has been done. It is well-known that this point is of crucial importance for diagnostic purposes, since the results of damage detection methods based on dynamic data strictly depend on the accuracy of the numerical model used to describe the reference configuration of the system.

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0.18

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0

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3L/4

L

-0.18

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-0.18

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Experimental mode

0

L/4

L/2

3L/4

L

0

L/4

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3L/4

L

0

L/4

L/2

3L/4

L

Analytical mode

Fig. 8. Reference configuration: comparison between experimental (EMA) and analytical (FEA, Model 1) mode shape of the normalized Modes 1 (column (a)) and 2 (column (b)), for upstream beam (top row), central beam (central row) and downstream beam (bottom row). Abutment: x¼ 0; pier: x¼L.

4.4. Calibration based on dynamic data Throughout the identification, the attention will be mainly focussed on the determination of a baseline model able to correctly reproduce the experimental behavior of the first two vibration modes of the bridge. There are at least three important reasons supporting this choice. First, experimental data turned out to be more reliable for the lower two modes, see, for example, the anomalous jump of the fourth EMA frequency from D3 to D4 shown in Fig. 3. Second, it is reasonably expected that lower modes are less affected by modeling errors, since numerical models usually loose accuracy by increasing the vibration mode order. Finally, the dynamic behavior of the bridge in the range of interest of seismic analysis is mainly determined by the lower vibration modes. Two main modeling issues were investigated in the reference configuration. The first issue concerns with the effect of existing degradation (i.e., cracking) on the dynamic response of the bridge. The second aspect involves a more accurate description of the boundary conditions. 4.4.1. Identification of existing degradation It is well-known that one of the most important effects impacting the modal parameters of a RC structure appears to be the cracking phenomenon that naturally occurs under structural weight. The crack pattern of the reference configuration depends on the mechanical properties of the material and, ultimately, on the static behavior of the whole bridge deck under dead loads. An accurate description of the existing pattern would require a large number of parameters and a deep non-linear analysis [15], which is outside the goals of the present research. Moreover, the analysis of the small vibrations in a neighborhood of the reference (cracked) configuration is a challenging topic still debated in the specialized literature. Here, to make the analysis significant for applications on large RC structures, such as bridges, and to make at the same time the interpretation model simple, the approach based on damage parametrization proposed in [24] is adopted, see also [3]. In a first stage, the effect of the cracking in the bridge deck is simulated as a decrease of the effective bending stiffness (Ec Ic ) of the longitudinal beams (having rectangular cross-section 0.35 m  1.20 m) through a reduction in the Young’s modulus Ec of the concrete. These assumptions are rather crude, but have the merit to maintain the linearity of the

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0.18

0.18

0.00

0.00

-0.18

0

L/4

L/2

3L/4

L

-0.18

0.18

0.18

0.00

0.00

-0.18

0

L/4

L/2

3L/4

L

-0.18

0.18

0.18

0.00

0.00

-0.18

0

L/4

L/2

3L/4

L

-0.18

2999

0

L/4

L/2

3L/4

L

0

L/4

L/2

3L/4

L

0

L/4

L/2

3L/4

L

Experimental mode

Analytical mode

Fig. 9. Reference configuration: comparison between experimental (EMA) and analytical (FEA, Model 1) mode shape of the normalized Modes 3 (column (a)) and 4 (column (b)), for upstream beam (top row), central beam (central row) and downstream beam (bottom row). Abutment: x¼0; pier: x¼ L.

dynamical problem. More precisely, following [24], it is assumed that the effect of cracking on a single longitudinal beam under dead loads causes a variation of the Young’s modulus of the concrete given by the expression

ðxÞ ¼ Ecrack c

8 > uncrack > ð1ð1aÞcos2 tðxÞÞ, > < Ec > uncrack > > , : Ec

  bL bL , x2  , 2 2     L bL bL L [ , , x 2  , 2 2 2 2

ð1Þ

see Fig. 10. In Eq. (1), tðxÞ ¼ px=bL,x 2 ½L=2,L=2 is the normalized distance along the beam axis measured from mid-span. The parameter b, 0 r b r1, defines the length of the cracked region of the beam. The parameter a, 0 r a r1, determines the magnitude of the degradation due to cracking, e.g., at mid-span Ecrack ð0Þ ¼ aEuncrack . Severe levels of cracking correspond c c uncrack to values of a close to 0. If a ¼ 1, no degradation is present. Ec denotes the Young’s modulus of the uncracked region of the beams. It should be noted that a symmetric cracking pattern is assumed by the symmetry of the bridge deck. In the present application, the extension of the cracked region is taken coincident with the interval of the beam span in which the bending moment of the deck cross-section is bigger than the corresponding first cracking bending moment Mcrack. The quantity Mcrack has been evaluated by considering the global cross-section of the deck, with 12 steel bars of 24 mm in diameter at the bottom of each longitudinal beam, and with steel bars in compression (16 mm in diameter) in the slab as shown in Fig. 11. Concrete belonging to the class C28=35 (Rck ¼ 35 N=mm2 ) and steel with yielding stress fyk ¼ 375 N=mm2 are used. Calculations show that M crack =Mmax C0:7, where Mmax is the maximum value of the bending moment due to dead loads evaluated at mid-span. The cracked region coincides approximately with the interval between the quarters of the beam span, e.g., b ¼ 0:5. Under the assumption that the beams have the same level of degradation, a preliminary estimate of a has been deduced by using the results by Wahab et al. [24] on static tests on simply supported RC beams, particularly beam 3 of [24] loaded by two equal forces applied at one-third and two-third of the beam length. The damage pattern shown in Fig. 9 of [24] suggests to take a C0:75 for our given ratio M crack =Mmax C0:7.

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M. Dilena et al. / Mechanical Systems and Signal Processing 25 (2011) 2990–3009

crack

Ec

uncrack

Ec 1

α

x L 2

βL 2

βL 2

0

L 2

Fig. 10. Damage function given by Eq. (1).

1∅16 /10

12 ∅24

12 ∅24

12 ∅24

x

0

L

M crack βL L Fig. 11. (a) Detail of reinforcement of the bridge deck; (b) determination of the cracked region of the longitudinal beams in the reference configuration of the bridge due to dead loads.

Model 2 was defined with b ¼ 0:5 and a ¼ 0:75. A comparison between EMA and FEA natural frequencies and corresponding vibration modes is presented in Table 3 and Fig. 12, respectively. Frequency reductions from Model 1 are appreciable, whereas, on the contrary, mode shape changes are almost negligible. As a consequence, Model 2 is not able to explain the non-symmetric behavior measured in experiments for the first two vibration modes. In order to improve Model 2, an extensive series of numerical simulations has been conducted by varying the severeness of the damage on the three longitudinal beams. In summary, the analysis showed that even significant changes in a were not able to justify the loss of symmetry of the mode shapes. Further analysis, not reported here for brevity, allowed to exclude the influence of the compliance of pile or abutment foundation. Taken together, these studies suggest to include the identification of the deck slab in the analysis. Until now, it was assumed that the slab was not affected by significant cracking as it was essentially subjected to compression under static vertical loads. However, as it was mentioned above, the bridge suffered of an exceptional flood on 2003 and, during that event (see Fig. 2), deck structures were subject to significant horizontal forces acting from upstream to downstream, with possible occurrence of tensile stress inside the downstream deck region due to in-plane bending in the horizontal plane. To take this effect into account, in Model 3 deck structures were divided into three main groups, one for each main beam, according to the scheme shown in Fig. 13. For the central beam and the deck slab relating to it, the stiffness coefficient

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L

0

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Model 3

Fig. 12. Reference configuration: comparison between experimental (EMA) and analytical (FEA, Models 1–3) mode shape of the normalized vibration Modes 1 (column (a)) and 2 (column (b)), for upstream beam (top row), central beam (central row) and downstream beam (bottom row). Abutment: x ¼0; pier: x¼ L.

Pier

Abutment

Downstream

Upstream Fig. 13. Identification of deck structure: groups of finite elements.

(e.g., Young’s modulus) was assumed as in Model 2. Instead, a stiffness multiplier was introduced for the portion of deck pertaining to upstream beam (du ) and downstream beam (dd ). Young’s modulus of the transverse beams was assumed constant and equal to 32 GPa. Since the previous results suggested that Model 2 underestimates the flexibility of downstream beam, the optimal values of the parameters du and dd were searched in the interval (0:95,1:20) and (0.80,1.00), respectively. These values were determined by minimizing the Euclidean distance ferr ¼ ferr ðdu , dd Þ between the experimental and analytical modal components of the first two vibration modes (evaluated at the measurement points,

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supports excluded), precisely ferr ðdu , dd Þ ¼

2 X N  X

2 uEMAðnÞ ujFEAðnÞ ðdu , dd Þ , j

ð2Þ

n¼1j¼1

and ujFEAðnÞ are the jth component of the nth experimental and numerical mode shape, respectively. A plot of where uEMAðnÞ j the error function is shown in Fig. 14. It turns out that the stiffness of the upstream region is underestimated by 10%, whereas that of the downstream region is overestimated by 20%. The updated model, called Model 3, accurately describes the first two modes of the bridge and reproduces the non-symmetry in the transverse direction detected in the experiments, see Fig. 12. The comparison also improves for higher modes, although significant differences still remain. With the exception of the third mode, natural frequencies are slightly underestimated, see Table 4. 4.4.2. Improvement of boundary condition The second modeling issue concerns with the description of the constraints at the end of the main beams on the pier side. This point is of importance, since it can be shown that the conversion of bearings from nominal simple supports (allowing displacements in the longitudinal direction) to nominal full fixity (i.e., pinned boundary conditions side pier) produces a significant increase in the fundamental frequency (from 8.763 to 13.187 Hz, for Model 3) and an increase, in lower percentage, in higher frequencies, see also [1]. Here, in order to obtain a more realistic description between the sliding constraint and the fixed constraint, in Model 4 each point support was modeled by inserting a linear elastic spring, of stiffness K, acting along the longitudinal direction of the bridge on each of the five nodes defining the constraint at the end of each beam on the pier side, see Figs. 6 and 15. Equal values for K are prescribed for the three beam supports. A sensitivity analysis carried out for K ranging on the interval [0,500] KN/mm shows that the fundamental mode is more sensitive to K and that changes are almost negligible on higher modes, see Fig. 15. A comparison between experimental and analytical frequency values of the updated model, Model 4, is presented in Table 4. The optimal value K OPT ¼ 15,000 N=mm was determined by matching experimental and numerical frequency of the first vibration mode. It can be shown that mode shapes of Model 3 are almost insensitive to the boundary conditions change. Finally, Model 4 is assumed as the analytical model of the reference configuration of the bridge.

f err

1.00 0.95

δd

0.90 0.85 0.80 0.95

1.00

1.05

1.10

1.20

1.15

1.25

δu

Fig. 14. Identification of the stiffness multipliers du and dd : plot of the error function given by Eq. (2).

Table 4 Reference configuration: comparison between experimental (EMA) and analytical (FEA, Models 3 and 4) values of natural frequencies pn (in Hz) and MAC EMA values. Dn ¼ 100  ðpFEA Þ=pEMA . n pn n EMA order

1 2 3 4 5

(Hz) pEMA n

9.430 13.487 26.218 34.764 47.171

Model 3

Model 4

FEA order

(Hz) pFEA n

Dn (%)

MAC

FEA order

(Hz) pFEA n

Dn (%)

MAC

1 2 4 5 7

8.763 13.270 27.430 33.558 43.950

 7.1  1.6 4.6  3.5  6.8

99.8 98.3 87.3 79.6 92.0

1 2 4 5 7

9.333 13.339 27.553 33.684 44.001

 1.0  1.1 5.1  3.1  6.7

99.8 98.4 88.8 82.7 89.5

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3003

30

f [Hz]

20

10 5K

0

0

100 1st frequency

200 300 K [kN / mm] 2nd frequency

400

500

4th frequency

Fig. 15. (a) Detail of the elastic constraint side pier; (b) sensitivity of the lower frequencies to K.

Fig. 16. Local refinement of the FE mesh of Model 4 near damage D1.

5. Structural identification of the damage configurations 5.1. Strategy of the identification The extreme variability of strain and stress fields in the region close to damage required a local refinement of the FE mesh of Model 4 of the reference configuration of the bridge. The solution that has been chosen is shown in Fig. 16. In a portion of the downstream beam that extends symmetrically 0.90 m beyond the external notches, shell FEs approximately square in shape and 0.05 m in side were inserted. This refined mesh has been suitably connected with the existing mesh of Model 4. Each notch was initially modeled by making independent the degrees of freedom (displacement and rotation) of the nodes of the FE mesh located on the two sides of the notch. This purely geometrical description of the damage turned out to be not accurate enough. In fact, the interruption of the main reinforcement bars leads to a substantial change in local stress in downstream beam as the bending moment in the damaged section vanishes. It follows that a redistribution of stresses in the bridge deck takes place inducing an enlargement of the cracked region and an increase of the overall compliance of the remaining two beams. The intensity of this effect varies from configuration to configuration, and this variability, as we will see below, it was taken into account. It should be noticed that the superposition of the above two effects leads to lower natural frequencies by well-known theorems of monotonicity for elastic systems, thus making inexplicable the increase in first two natural frequencies measured in experiments from configuration D3 to D5 (see Fig. 3). It was therefore necessary to introduce new elements to justify the increase in frequency, and particularly in the fundamental frequency. As it will be explained shortly, these elements relate to a sort of stiffening of the boundary conditions side pier induced by the change in structural behavior of the deck during the evolution of the damage. 5.2. Reconstructing damage evolution Damage configurations D1 and D2 have been modeled by simply making independent the degrees of freedom of nodes of the FE mesh positioned on both sides of the notch. The change of the tension field caused by the interruption of the

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lower bars of the beam was not considered in modeling at this stage. The evolution of the first natural frequencies and associated mode shapes to damage are shown in Table 5, Figs. 17 and 18, respectively. At first, even the damage D3 was modeled in the same way. However, numerical simulations have shown that this would lead to frequency reductions extremely lower than the corresponding experimental ones. In fact, due to vanishing of the bending moment in the adjacent section (damage D2), notch D3 is located in a portion of the beam with negligible stress state. For this reason, damage configuration D3 has been determined by considering both the additional cracking induced by the first levels of damage D1–D2 and the geometrical discontinuity caused by the notch corresponding to damage D3. The cracking induced by damage D1–D2 is simulated, as before, as a decrease of the effective bending stiffness of the longitudinal beams and it is superimposed to the existing degradation of the reference configuration. The extension of the cracked region in each structural element is taken coincident with the interval of the span length in which the bending moment is bigger than the first cracking moment of the transversal cross-section. The bending moment distribution has been evaluated by modeling the bridge deck as a beam grillage, with a hinge at the notched cross-section (damage D1–D2) of the downstream beam. By following the same procedure illustrated in Section 4.4.1, the parameter a

Table 5 Evolution of the first natural frequencies to damage predicted by the FE models of the bridge. Configuration

EMA order

(Hz) pEMA n

FEA order

pFEA (Hz) n

Dn (%)

MAC

Damage D1

1 2 3 4 5

9.382 13.520 25.834 34.535 47.467

1 2 4 5 7

9.253 13.272 27.126 33.197 43.958

 1.4  1.8 5.0  3.9  7.4

99.8 98.6 82.2 74.0 91.6

Damage D2

1 2 3 4 5

9.215 13.402 24.897 34.340 47.251

1 2 4 5 7

9.036 13.142 26.832 32.493 43.597

 1.9  1.9 7.8  5.4  7.7

99.8 98.4 73.4 69.1 92.5

Damage D3

1 2 3 4 5

8.994 13.190 24.650 33.532 46.786

1 2 4 5 7

8.959 13.074 26.694 32.356 43.515

 0.4  0.9 8.3  3.5  7.0

99.7 98.5 74.7 56.1 91.9

Damage D4

1 2 3 4 5

8.970 13.247 25.096 35.283 47.151

1 2 4 5 7

8.852 13.013 26.682 32.313 43.410

 1.3  1.8 6.3  8.4  7.9

99.8 98.3 75.8 72.1 90.3

Damage D5

1 2 3 4 5

9.092 13.329 25.288 35.230 46.800

1 2 4 5 7

8.960 13.013 26.693 32.341 43.408

 1.5  2.4 5.6  8.2  7.3

99.8 98.4 73.5 76.1 89.0

Damage D6

1 2 3 4 5

9.030 13.284 25.860 35.094 46.722

1 2 4 5 7

8.858 12.965 26.678 32.311 43.368

 1.9  2.4 3.2  7.9  7.2

99.8 96.8 74.3 75.9 85.2

9.6

13.6

9.4

13.4

9.2

13.2

9.0

13.0

8.8

R

D1

D2

D3

D4

D5

D6

12.8

Experimental frequency

R

D1

D2

D3

D4

D5

D6

Analytical frequency

Fig. 17. Evolution of the lower experimental and analytical frequencies to damage: (a) Mode 1; (b) Mode 2.

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Damage D3

Damage D5

Fig. 18. Evolution of the mode shape of the normalized vibration Modes 1 (column (a)) and 2 (column (b)) to damage predicted by the FE models of the bridge, for upstream beam (top row), central beam (central row) and downstream beam (bottom row). Abutment: x ¼0; pier: x¼ L.

Pier

Abutment

Downstream

Upstream Fig. 19. Evolution of the cracked region in longitudinal beams. Light grey: cracked region in the reference configuration; dark grey: extension of the cracked region due to damages D1 and D2.

turns out to be almost unchanged with respect to the value attained at the reference configuration. On the contrary, the cracked area is larger in the central beam and, to a lesser extent, in the upstream beam, see Fig. 19. Transverse beams turn out to be uncracked. The regions that were already cracked in the referential configuration were maintained cracked in the subsequent damage configurations. Damage configuration D4 has been described by extending the depth of the notch D3 to the whole transversal crosssection of the beam. The cracking distribution was assumed to be coincident with that of damage D3. Changes in boundary conditions will now be examined. The trend of the vertical support reaction of each longitudinal beam (evaluated under dead loads) to the evolution of the damage is shown in Table 6. The values side abutment are substantially unchanged. Values corresponding to downstream and upstream beam tend to decrease on the pier side.

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Table 6 Evolution of the global vertical support reaction of each longitudinal beam (evaluated under dead loads) with respect to damage. Values in kN. Position

Configuration Ref.

D1

D2

D3

D4

D5

D6

Abutment Upstream Central Downstream

155 162 162

155 162 162

155 162 161

156 165 157

155 166 157

155 166 157

154 166 159

Pier Upstream Central Downstream

158 168 152

154 178 147

143 200 136

139 205 136

137 209 133

137 209 133

139 209 132

On the contrary, the increase of the vertical reaction in the central beam is significant (about 25%). It occurs mainly in the transition from D1 to D2. These calculations refer to the 2D model of the bridge, in which the damage was described only as a geometrical discontinuity (notch) in the downstream beam. One can show that similar conclusions are reached taking into account the evolution of the cracking caused by the redistribution of stress in the bridge deck. It is reasonable to assume that increase in vertical reaction leads to accentuation of local friction phenomena between the surfaces in contact between the central beam and the pier. Within the limits of the macroscopic mechanical model of supports that we have adopted, this effect may be described as an increase in the stiffness K of elastic link elements that contrast longitudinal movement of the point supports of the central beam. The value of K was estimated by matching the theoretical and experimental value of the fundamental frequency in D5 configuration. More specifically, the percentage error between experimental and analytical value was imposed to be the same as in damage configuration D4. The updated value of K is about 25,000 N/mm. Finally, damage configuration D6 has been described by extending the depth of the notch to the whole transversal cross-section and assuming K ¼25,000 N/mm. Table 5 summarizes the experimental and theoretical values of natural frequencies at each damage configuration. Concerning Modes 1 and 2, modeling errors are very small (less than 2.5% for D1–D6) and almost uniform for the various damage configurations. It is worth noticing that the FE model is able to reproduce the anomalous trend of the first and (in part) of the second natural frequency, see Fig. 17. Corresponding mode shapes are described with high precision by the FE model. More importantly, evolution of the mode shape with respect to damage is accurately reproduced, as it can be deduced by comparing Figs. 4 and 18. However, it should be remarked that the FE model is not able to explain the behavior of higher vibration modes measured during experiments. Here, discrepancies are still large and difficult to justify. 6. Damage localization based on modal curvature changes In this section, the results of a damage localization method based on the determination of changes in curvature of the first vibration modes of Dogna Bridge are presented. It is well-known that bending modal curvature can be considered as a good indicator of localized damage in beam structures. This diagnostic method was originally proposed in [16] for damage localization in beams. Applications to fullscale bridge structures are rare and are hampered by a number of difficulties [23]. Primarily, an accurate estimate of the second derivative of the mode shape is needed. Another source of uncertainty is connected with the diffuse character of the damage. Finally, it seems no so easy to transfer the method to structural models less schematic than beams, such as plates or combination of plates and beams, as in the present case. Despite these difficulties, it was shown in [5] that if the working hypotheses on which the modal curvature method is based are satisfied (e.g., concentrated damage, predominantly 1D behavior, accurate curvature determination), then changes in modal curvature of the first lower modes can be successfully used to identify the location of the damage. Here, we follow the approach presented in [5] and we refer to that paper for a detailed description of the procedure. In brief, denote by 0 ¼ x0 o x1 o    o xN ¼ L the measurement points along a given longitudinal beam. This decomposition of the interval ½0,L will be denoted by d. A cubic spline function fd associated with the decomposition d is a function fd : ½0,L-R, such that fd 2 C 2 ð½0,LÞ and coincides with a third order polynomial on every subinterval ½xi ,xi þ 1 , i ¼ 0,1, . . . ,N1. Here, C 2 ð½0,LÞ is the set of continuous functions with continuous first and second derivative on ½0,L. ðrÞ Let yðrÞ ¼ ðy0ðrÞ ,y1ðrÞ , . . . ,yN Þ be the restriction of the rth normalized mode of the bridge to the set of measurement points belonging to the beam. A cubic spline function fd ðyðrÞ ; Þ defined on ½0,L and such that fd ðyðrÞ ; xi Þ ¼ yðrÞ i ,

i ¼ 0,1, . . . ,N

ð3Þ

is called natural cubic spline function if f 00d ðyðrÞ ; x0 Þ ¼ 0 ¼ f 00d ðyðrÞ ; xL Þ:

ð4Þ

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0.02

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0.01

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0.02

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3007

0

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-0.02

0

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L

0

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L

0

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3L/4

L

Fig. 20. Variation of the modal curvature of the first (normalized) experimental vibration mode from the reference configuration of the FE model (Model 4) to actual damage configuration for upstream beam (top row), central beam (central row) and downstream beam (bottom row). Column (a): damage D1; column (b): damage D3; column (c): damage D5. Abutment: x¼0; pier: x¼ L.

Note that these conditions are certainly verified at the ends of longitudinal beams because of the simple support end conditions. Under the above assumptions, it is possible to show that there exists a unique natural cubic spline function fd ðyðrÞ ; Þ associated to the decomposition d and data yðrÞ . An efficient algorithm for reconstructing the natural spline function is also available, see [5] for more details. Real applications often require to work without any experimental information on the reference configuration of the bridge. Therefore, changes in modal curvature can be reconstructed only from a numerical model of the reference configuration of the bridge. Figs. 20 and 21 show, by way of example, the variations of curvature of the first and second normalized vibration mode evaluated on the basis of the Model 4 of the bridge described in Section 4.4.2. Results clearly show a significant increase (in absolute value) of the modal curvature on the downstream beam, near the third-span/ fourth-span side pier. This indication is also confirmed by the appearance of changes, this time more limited, on the same region of the central beam, while the modal curvature of the upstream beam remains approximately constant for the various damaged configurations. Similar information, even if slightly less accurate, can be extracted from the third vibration mode. It should be recalled that similar results, not reported here for brevity, were obtained by using Model 1 as numerical model of the reference configuration of the bridge. This suggests that the proposed damage identification technique has a certain degree of stability even to rough descriptions of the reference configuration of the bridge. This point is of interest for applications, since in most practical cases the reference configuration of the system is not available. 7. Conclusion An interpretation of dynamic tests carried out on a single-span concrete bridge in its reference state and in a series of progressive damage configurations has been presented in this paper. The analysis has been mainly devoted to provide a justification of the natural frequency trend measured and to explain an asymmetric behavior of the fundamental mode shape detected during the experiments. It is shown that the definition of an accurate model of the bridge should take into account the increase in flexibility due to cracking of the deck structures

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0.02

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L

Fig. 21. Variation of the modal curvature of the second (normalized) experimental vibration mode from the reference configuration of the FE model (Model 4) to actual damage configuration for upstream beam (top row), central beam (central row) and downstream beam (bottom row). Column (a): damage D1; column (b): damage D3; column (c): damage D5. Abutment: x¼ 0; pier: x ¼L.

naturally induced by the dead loads and, in addition, the change in the boundary conditions at the ends of the main beams at the various damage levels. The last part of the paper has been addressed to damage localization by dynamic data. Changes in modal curvature of the first two modes evaluated along the main beams were successfully used to identify the location of the damage. The obtained results seem to confirm the importance of information on vibration modes for the localization of structural damage in full-scale bridges.

Acknowledgments This research was made possible thanks to the interest of the Dipartimento della Protezione Civile of the Friuli Venezia Giulia and the logistical support of the I.CO.P. Co. (Udine, Italy). The writers would like to gratefully acknowledge the cooperation of Drs. G. Berlasso and C. Garlatti. The collaboration of Drs. F. Alessandrini and A. Coccolo is gratefully appreciated. References [1] J.M.W. Brownjohn, P. Moyo, P. Omenzetter, L. Yong, Assessment of highway bridge upgrading by dynamic testing and finite-element model updating, Journal of Bridge Engineering ASCE 8 (3) (2003) 162–172. [2] F.N. Catbas, D.L. Brown, A.E. Aktan, Use of modal flexibility for damage detection and condition assessment: case studies and demonstrations on large structures, Journal of Structural Engineering ASCE 132 (2006) 1699–1712. [3] M.N. Cerri, F. Vestroni, Detection of damage in beams subjected to diffused cracking, Journal of Sound and Vibration 234 (2000) 259–276. [4] M. Dilena, A. Morassi, Reconstruction method for damage detection in beams based on natural frequency and antiresonant frequency measurements, Journal of Engineering Mechanics ASCE 136 (3) (2010) 329–344. [5] M. Dilena, A. Morassi, Dynamic testing of a damaged bridge, Mechanical Systems and Signal Processing 25 (5) (2011) 1485–1507. [7] C.R. Farrar, D.A. Jauregui, Comparative study of damage identification algorithms applied to a bridge: I. Experiment, Smart Materials and Structures 7 (1998) 704–719.

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