Proceedings of the IMAC-XXVII
February 9-12, 2009 Orlando, Florida USA
©2009 Society for Experimental Mechanics Inc.
Variation of modal parameter estimates of a prestressed concrete bridge
D. Siegert1 , L. Mevel2 , E. Reynders3 M. Goursat4 , G. De Roeck3
1
Université Paris-Est, LCPC, 58 boulevard Lefebvre 75732 Paris, France
2
3
4
IRISA/INRIA, Campus de Beaulieu 35042 Rennes, France
K.U. Leuven Structural Mechanics Division, Leuven, Belgium
INRIA, domaine de Voluceau-Rocquencourt 78153 Le Chesnay, France
Abstract
This paper presents the processing methods and the analysis results of ambient vibration data recorded during a sixmonth period on a highway bridge. Data analysis was focused on the variations of the modal parameter related to the
accuracy of the estimates and to the temperature effects. The first flexural and torsional modes were estimated with
their variance from short acceleration time series records using stochastic covariance driven subspace identification
techniques. The frequency variation estimates were compared to the variations induced by structural modifications
simulated with a finite element model for assessing the detection threshold level. Subsequently the temperature
induced variations in the measured frequencies were analysed. Results of a damage detection test based on the
computation of a null-space residual derived from the covariance estimates show the response to the temperature
effects. Significant increases in the variance of the parameter estimates were also detected by the test. Finally, the
effectiveness of the temperature robust version of this promising damage detection method was investigated with
the available data records.
1 INTRODUCTION
Vibration measurements of structures in ambient excitation conditions have been used as a means for structural
damage detection. Since the damage is not directly identified, this technique belongs to the so-called inferred
classes [1] . The detection of a change in the modal properties is related to a modification of the structural stiffness
which may be induced by a structural damage or by temperature effects.
Structural modification leading to a decrease of the resonant frequencies of 1% was successfully detected in laboratory conditions when the temperature was held fairly constant and for frequency estimates with a coefficient of
variation of 0.15% [2] . Encouraging experimental results were obtained also in laboratory under variable temperature conditions using empirical frequency corrections and statistical process control techniques [4] . The variability
of the resonant frequencies of a bridge induced by the environmental conditions was highlighted and quantified e.g.
in [3] . Frequency variations up to 7% were observed over a 24-hour monitoring period. It was also pointed out that
effective correlation models were difficult to obtain.
This paper focuses on the results of the ambient vibration monitoring of a bridge carried out during a five month
period. Acceleration data were processed using covariance driven stochastic subspace identification algorithms
(COV-SSI). Uncertainty bounds of the frequency estimates were calculated using the variance estimation procedure
presented in [5] . This procedure is based on the first order sensitivity of the modal parameter estimates to perturbations of the measured output-only data. As an alternative technique to frequency control charts, the computation of a
null space residual derived from the covariance estimates of the response data was proposed for detecting structural
damage [7][12][8][9] . This method was used with the measured data for investigating the temperature effects.
2 DESCRIPTION OF THE BRIDGE
The considered structure consists of five prestressed concrete isostatic girders connected by an overall concrete
deck and five cross braces as shown in Figure 1.
Figure 1: View of the superstructure.
The instrumented span is 33 m long, it carries three one way lanes and is part of a multi-span bridge. The deck is
connected to an expansion joint at the bank side and the top slabs at the other end are connected to ensure the
continuity of the pavement. This design allows the transmission of horizontal forces while leaving the beam rotations
free.The girder beams are supported by laminated rubber bearings.
The bridge is located on a heavy trafficked European itinerary in the North of Paris and undergoes a free-flowing
traffic. The mean traffic flow during the working days was about 10000 trucks/day for heavy trucks with a gross
vehicle weight over 35 kN.
3 AMBIENT VIBRATION TEST CONDITIONS
The tests were conducted during the period from August to December 2004. The ambient traffic-induced vibration
data were measured in the vertical direction with inductive accelerometers HBM B12-200 and the Spider 8 HBM
acquisition system. A photography of an instrumented girder is shown in Figure 2. The measurements were remotely
triggered and the data transfered from the personal computer on the field to the computer at the office.
Figure 2: Instrumented girder.
Most often, only one accelerometer was used for the monitoring. It was located at mid-span on the side girder under
the more heavily trafficked lane. An amount of 42 signals of 25 minutes duration with a 50 Hz sampling frequency
were gathered for the post-processing. A typical time-history of the measured acceleration is shown in Figure 3.
Acceleration data with a good signal to noise ratio were produced by heavy vehicles circulating on the bridge.
The ambient temperature was measured during the monitoring period with a data logger, placed under the bridge,
at 30 minutes intervals. The temperature recorded on the site from the summer to the winter season. The daily
variations were relatively small compared to the seasonal variation which ranged from 23 ◦ C to 3 ◦ C.
The setup with 7 roving accelerometers and the reference accelerometer A1 located at mid-span, was used for the
mode shapes identification. The accelerometer setup with 15 different positions on the girders is shown in Figure 4.
0.3
acceleration (m/s²)
0.2
0.1
0.0
−0.1
−0.2
−0.3
0
20
40
60
80
100
120
140
160
180
200
time (s)
Figure 3: Time-history measured at mid-span during ambient vibration tests.
E2
12.4 m
E1
E3
A8
A7
A6
A5
A4
A3
A2
A8
A7
A6
A5
A4
E4
E5
P5
P4
P3
y
3.10 m
P2
x
P1
A2
A3
A1
8.25 m
33 m
Figure 4: Accelerometer locations.
4 PROCESSING METHODS
4.1 Covariance driven Subspace System Identification Method
The modal identification algorithms use the covariance driven subspace methods which are described in full detail
in [10] [11] . Here, we just present some of the definitions and basic computational steps, useful to set the parameters
of the method for processing the measurement data of free vibrations to identify the linear time invariant system.
The linear state-space difference equations of the corresponding model are given by
Xk+1
=
F Xk + Vk+1
Yk
=
HXk + Uk
where Xk denotes the state vector of the system sampled with a frequency Fs = τ1 and Y (k) the vector of the
measured output responses. Vk and Uk are respectively the white noise excitation process and the white noise
in the measurement. The identification procedure of a linear time invariant system is based on the covariance
estimates Ri which are defined as
Ri =
N
X
1
T
Yk Yk−i
(N − i)
k=i+1
T
where the upper script is the transpose operator, N the number of recorded samples and i the time lag of the
covariance. The Hankel matrix of size (p + 1)r × (p + 1)r is then written as
2
6 R0
Hp+1,p+1
6
6
6 R1
=6
6 ..
6 .
4
Rp
R1
R2
..
.
Rp+1
..
.
..
.
..
.
..
.
3
Rp
Rp+1
..
.
7
7
7
7
7
7
7
5
R2p
where r is the number of sensors or measurement channels selected for the identification data processing. The
identification of the state transition matrix F of the state-space model of the system rely on the factorization of the
Hankel matrix which is derived from the state-space equations
Hp+1,p+1 = Op+1 (H, F )Cp+1 (F, G)
T
and C is the controllability
where O is the observability matrix defined as Op+1 (H, F ) = H HF · · · HF p
matrix defined as Cp+1 (F, G) = G F G · · · F p in which G = E(Xk YkT ) is the covariance matrix between
the state vector and the vector of the measured output responses. Only the observability matrix needs to be
identified using the singular value decomposition of the Hankel matrix. Then the state transition matrix F is obtained
from the least squares solution of the over determinated linear system
Op+1 (r + 1 : r(p + 1), :) = Op F
where the symbol : denotes a vector of integer subscripts.
4.2 Null-space residual monitoring technique
The eigenanalysis of the state transition matrix leads to define the vector θ0 whose components correspond to the
eigenvalues and the observed components of the eigenvectors stacked in a single column vector. The left kernel of
both the observability matrix and the Hankel matrix is defined as
S T (θ0 )Hp+1,p+1 = S T (θ0 )Op+1 = 0
where the rows of S T (θ0 ) form an orthonormal basis of the left null-space of the observability matrix of the system
in the reference state identified from the Hankel matrix. Then the residual single column vector for the reference
state θ0 is
√
ζN (θ0 ) = N vec(S T (θ0 )Hp+1,p+1 )
and the value of the test corresponds to the χ2 variable defined as
T −1
χ2N = ζN
Σ̂ ζN
T
where Σ(θ0 ) = E(ζN ζN
) is the covariance matrix of the residual. The monitoring of the system consists in calculating the value of the test with the Hankel matrices estimated from the new recorded output data. A significant
increase in the χ2 value indicates that the system is no more in the reference state.
Since the test may be sensitive to environmental temperature variations, a robust version of the null-space based
method has been introduced in reference [9] for rejecting the temperature effects. It consists in determining the left
null-space of the average of Hankel matrices over various temperature conditions of the undamaged structure. The
set of temperature conditions taken into account for defining the reference state must be large enough to be likely
to include the temperature conditions of the damaged structure.
4.3 Variance estimation procedure of the modal parameters
Since in reality the assumption underlying the COV-SSI algorithm are only fulfilled approximately (infinite amount
of data, stationary white process and measurement noise, linear system, known true model order, ...), there is a
stochastic uncertainty on the identified system matrices (F, H), resulting in a bias error and a variance error on
the modal parameters obtained from the identification method. The biais error is usually partially removed using
the stabilization diagram. The variance error cannot be removed, but it can be estimated using a linear sensitivity
analysis as demonstrated in [5] .
If xi and yi are the right and left eigenvectors corresponding to an eigenvalue λi of the estimated F matrix, the
covariances of the discrete-time system poles λi and λj can be estimated as
cov(<(λi ), <(λj )) = <(
(xi ⊗ yj )T
(xi ⊗ yj )T T
)cov(vec(F ))<(
)
∗
yi xi
yi∗ xi
where x denotes the complex conjugate of x and x∗ denotes the complex conjugate transpose of x, the real part
of the matrix is stacked on top of the imaginary part and vec is the column stacking operator. A validate formula for
cov(vec(F )) is provided in [5] .
From the covariance expression of the discrete-time system poles, the covariances of the eigenfrequencies and
damping ratios can be obtained. The eigenfrequencies fi and the damping ratios ξi are obtained after converting
the discrete time poles to the continuous time poles using the following formulas
ln λi
∆T
|λci |
fi =
2π
λci =
<(λci )
|λci |
Since these estimates are usually quite accurate, a linear sensitivity analysis is sufficient for obtaining the covariances of the eigenfrequencies fi and the damping ratios ξi
ξi =
cov(<(λci ), <(λcj )) = Jλ cov(<(λi ), <(λj ))JλT
cov(fi , ξj ) = Jf cov(<(λci ), <(λcj ))JξT
Where Jλ , Jf and Jξ are linear sensitivity matrices, for which explicit expressions are provided in [5] .
5 RESULTS
5.1 Experimental modal identification
The signals were processed with the COSMAD toolbox developed under the Scilab software to extract the modal
parameters of the bridge deck. Figure 5 shows the frequency stabilisation diagram where the frequencies in Hertz
are plotted against the order of the subspace method. Two Modes of vibration with eigenfrequencies close to 4 Hz
were identified. The damping coefficients are within the range from 2% to 4%. The mode shapes are plotted in
Figure 6, the bending mode is associated with the frequency 3.9 Hz and the torsion mode with 4.3 Hz.
Stabilization diagram
5.0
4.5
4.0
3.5
Frequency
3.0
2.5
2.0
1.5
1.0
0.5
0.0
10
20
30
40
50
60
Model order
Figure 5: Frequency stabilisation diagram.
70
35
1.0
0.8
0
0.6
1.00
30
5
25
Z
0.4
0.40
20
Z
10
0.2
15
−0.20
0.0
15
20
0
2
25
4
Y
6
8
30
10
12
35
14
X
10
−0.80
14
12
X
5
10
Y
8
6
4
2
0
Figure 6: Shapes of the bending mode (3.9 Hz), torsion mode (4.3 Hz)
5.2 Resonant frequency monitoring
Figure 7 shows the correlations of the two eigenfrequencies with the measured temperature. The decrease of
frequencies with temperature is only observed for the bending mode. In this case, the relationship is fairly linear
with a slope of 0.005 Hz/◦ C. The variations of the torsion mode frequency are of the same magnitude but a linear
correlation with the temperature was not observed.
The measured modal frequency changes might be explained by the temperature sensitivity of the bearing device.
Although, the frequency changes of the torsion mode cannot be fully explained by this temperature effect.
Large simulated damages with a finite element model, i.e. a reduction of 30% of the bending stiffness over 1 m at
mid-span of two girders, produce frequency shifts close to 1% [6] .
The 95 % confidence interval estimates of the frequencies were calculated with the method presented in section
4.3 and are plotted in Fig. 8. The mean value of the coefficient of variation is close to 0.5 % except for two samples
with higher values for the torsion mode. The coefficient of variation of the residual in the linear fit of the bending
frequency is also close to 0.5 %.
5.3 Null-space residual monitoring
The test values were first calculated with a vibration signature measured in summer. The results are displayed in
Fig. 9 where the sample order follows the chronological order of the measurements. Dramatic increases of the test
values are related to the samples with very large values of uncertainties and to the significant change of the bending
frequency induced by the seasonal variation of temperature. The deviation of the test corresponds to a frequency
modification of about 4 times the uncertainty bound.
Fig. 10 shows the time series of the χ2 test values obtained for a vibration signature mixing vibration responses
measured at different temperatures. The test is no more sensitive to the seasonal temperature changes. The
observed deviation corresponds to the record with a high value of uncertainty in the frequency of the torsion mode.
4.5
jjLS fit
bending
bending
torsion
4.4
torsion
ls−fit
Frequency (Hz)
4.3
4.2
4.1
4.0
3.9
3.8
0
5
10
15
20
25
Temperature (°C)
Figure 7: Correlation with the measured temperature for the frequency estimates
eigenfrequency [Hz]
5
4.5
4
3.5
0
5
10
15
20
25
temperature [oC]
Figure 8: 95 % confidence interval estimates (2σ) for the frequencies
350
300
Test value
250
200
150
100
50
0
0
10
20
30
40
50
60
70
order of the samples
Figure 9: Time series of the χ2 test values for a signature recorded in summer.
7
6
Test Value
5
4
3
2
1
0
0
5
10
15
20
25
30
35
Order of the samples
Figure 10: Time series of the χ2 test values for a signature with mixed temperature scenarios.
6 CONCLUSIONS
Ambient vibration data of a bridge undergoing a heavy traffic were measured and processed for analysing the
variations of the modal properties of the tested structure. The random variations of the frequency estimates were
calculated for each measured vibration data sample. Except a few data records, the coefficient of variation was
about 0.5 %. This value is much higher than in laboratory conditions where the coefficient of variation ranges from
0.05 % to 0.15 % when free vibration responses can be collected with a high signal to noise ratio.
Modal property changes induced by temperature variations were also investigated. In our case the environmental
effect on the frequency of the torsion mode was not described by a simple linear correlation. Frequency shifts in the
range of 2 % was observed during the monitoring period. Application of the χ2 test has shown that a modification
was detected when significant shifts of the temperature were reached i.e 3 times the standard deviation of the
frequency. The test was also sensitive to an increase of the standard deviation of the frequency estimate. The
effectiveness of the temperature robust version of the test with the use of signatures including several temperature
conditions was confirmed.
ACKNOWLEDGMENTS
The authors acknowledge F. Marc and V. Le Cam for their contribution in setting the remotely data transfer system.
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