Analysis and Extraction of Temperature Effects on Natural
Frequencies of a Footbridge based on Continuous Dynamic
Monitoring
Wei-Hua Hu, Carlos Moutinho, Filipe Magalhães, Elsa Caetano & Álvaro Cunha
Faculty of Engineering of University of Porto (FEUP), Portugal
ABSTRACT: The development of efficient vibration based structural health monitoring
systems requires distinguishing between abnormal changes in modal parameters caused by
structural damage and normal changes due to varying environmental conditions. In this context,
this paper is focused on the analysis and extraction of effects of temperature oscillations on
natural frequencies of a footbridge, where a long-term dynamic monitoring system was installed
by the Laboratory of Vibrations and Monitoring of the Faculty of Engineering of University of
Porto. With that purpose, firstly, the temperature influence on the natural frequencies is
reported, and correlations between measured temperatures and estimated natural frequencies are
analyzed. Then, the Principal Component Analysis (PCA) and the Novelty Detection method
are applied to identified natural frequencies: PCA effectively eliminates environmental
influence; Novelty analysis on the residual error of PCA predicted model is used as a statistical
indication of damage. The proposed procedure is illustrated using continuous dynamic data
collected from the footbridge during more than one year.
1 INTRODUCTION
Structural Health Monitoring (SHM) has become a major international research topic in recent
years in civil engineering. One of the main obstacles in the application of SHM is the
environmental and operational variations of structures. The so called ‘damage sensitive’
features are also sensitive to changes in environmental and operational conditions of structures,
which often mask subtle structural changes caused by damage (Hoon Sohn 2007). Therefore,
important issues in SHM are the analysis of influences of environmental and operational
variations, and the achievement of damage indication based on the removal of such effects.
In this context, this paper is focused on the analysis and extraction of effects of
environmental and operational conditions on natural frequencies of a slender footbridge, where
a long-term dynamic monitoring system was installed by the Laboratory of Vibrations and
Monitoring (ViBest, www.fe.up.pt/vibest ) of FEUP. Firstly, the temperature influence on
natural frequencies is reported, and correlations between measured temperatures and estimated
natural frequencies are analyzed. Then, the Principal Component Analysis and the Novelty
Detection method are applied to identified natural frequencies. The proposed procedure is
illustrated using continuous dynamic data collected from the new Coimbra footbridge.
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2 DESCRIPTION OF THE BRIDGE AND DYNAMIC MONITORING SYSTEM
The new “Pedro e Inês” footbridge over Mondego River is located in the centre of the City Park
of Coimbra, recently developed along the two banks of the river and opened to public in April
2007. This new infrastructure, conceived to become a landmark for the city and to contribute to
the quality of a new leisure area, was designed by Adão da Fonseca (Adão da Fonseca et al
2005), leading a team from AFAssociados, in collaboration with Cecil Balmond, leading the
architectural team from Ove Arup. The bridge has a total length of 275m and is formed by a
parabolic central arch with a span of 110m and two half lateral arches, in steel, supporting with
total continuity a composite steel concrete deck (Fig.1). The anti-symmetry of both arch and
deck cross-sections along the longitudinal axis of the bridge is a unique feature of this bridge,
leading to the creation of a central square with 8m×8m at mid-span.
AV1
AV2
AV5
AV3
S3
30.5
0m
64.00
AV6
S2
55.00
m
S1
55.00
m
64.00
m
west
m
6.00
m
east
AT4
(a) Bridge plan and elevation, deployment of accelerometers
and sections (S1-S3) with temperature sensors
North
North
South
T2
TA South
T1
Section 1 (S1)
T3
TC
North
Section 2 (S2)
AT4
AV5
South
T4
Section 3 (S3)
(b) Sections (S1-S3) and temperature sensors
Figure 1. Deployment of accelerometers and temperature sensors
Numerical and experimental studies, developed by the Laboratory of Vibration and
Monitoring from FEUP, showed that this slender footbridge is prone to excessive vibrations
caused by groups or streams of pedestrians. Therefore, six groups of tuned mass dampers
(TMDs) were installed (Caetano, Cunha et al 2008).
Aiming the permanent characterization of vibration levels after construction, the footbridge
was also instrumented with a dynamic monitoring system, formed by signal acquisition, data
communication and signal processing modules. The signal acquisition system comprises six
uniaxial piezoelectric accelerometers installed in correspondence with the location of TMDs
(Fig. 1). Five of them measure vertical accelerations (AV1-AV3, AV5-AV6), whereas another
one measures lateral vibrations at mid-span (AT4). All sensors are mounted inside the metallic
deck and wired to the corresponding signal conditioners and digital computer incorporating an
analogue to digital converter and a UPS system, located in one of the concrete abutments of the
structure. An automatic signal acquisition toolkit was developed in LabVIEW environment to
record the acceleration signals and generate setup files every 20 minutes. The data
communication system sends permanently the most recent collected data to a computer located
at FEUP using an ADSL line (Moutinho et al 2008). The signal processing system is a toolkit
developed in LabVIEW. It automatically searches the latest data transmitted from the bridge in
57
Coimbra, detects maximum vibration amplitudes and makes statistical treatment of acceleration
time series, generates waterfall plots to depict the frequency component distribution and
identifies modal parameters using automated EFDD and SSI-COV techniques (Hu et al 2008).
Besides the accelerometers, 6 temperature sensors (Fig. 1) were also installed by the
Laboratory for Concrete Technology and Structural Behaviour of FEUP to provide
environmental data, including the ambient temperature (TA), concrete slab temperature (TC),
and temperature in different steel sections (T1-T4), which were well selected by the designer in
order to provide an adequate representation of the bridge behaviour (Dimande et al 2008).
The set of mentioned components can be considered as a simple dynamic structural health
monitoring system that has been operating since the 1st of June 2007 till now, except for a stop
from the 1st of September 2007 to the 13th of October 2007.
3 EFFECT OF TEMPERATURE ON NATURAL FREQUENCIES
In this study, the temperature data from the 6 temperature sensors exhibit similar trend and
therefore the average temperature is used to represent environmental variation. Fig. 2 depicts
annual variation of one-hour average temperature during day light time. It is observed that the
average temperature at the footbridge ranges from 0.19°C to 34.40°C.
Figure 2. Variation of one-hour average temperatures from 1st June 2007 to 31st May 2008
(The monitoring system was not operating from 1st Sep 2007 to 13th Oct 2007)
frequency
daily average frequency
(a) 1st natural frequency identified by EFDD
frequency
daily average frequency
(c) 1st natural frequency identified by SSI-COV
frequency
daily average frequency
(b) 2nd natural frequency identified by EFDD
frequency
daily average frequency
(d) 2nd natural frequency identified by SSI-COV
Figure 3. Variation of identified modal frequencies by both EFDD and SSI-COV methods from 1st June
2007 to 31st May 2008 (The monitoring system was not operating from 1st Sep 2007 to 13th Oct 2007)
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The structure natural frequencies have been automatically identified by the signal processing
system for each hour. Fig. 3 shows the annual variation of the first two natural frequencies
estimated by the used methods. To further illustrate the annual tendency, the daily average
frequency curve is also displayed. It is observed from Figs. 2-3 that the frequency domain
EFDD and the time domain SSI-COV methods produce similar results, and the first two natural
frequencies are inversely related to changes in measured temperature.
The relation between natural frequencies and temperature may be further clarified by Fig. 4.
As shown, the first two natural frequencies identified by different methods both decrease as
temperature rises. From a statistical point of view, the relations between temperature and
natural frequencies can be assumed as linear. Therefore, a linear regression model was
developed to represent the first two natural frequencies as function of the average temperature,
which can be mathematically described as fi=a+bti, where a and b are coefficients to be
estimated, ti are the samples of average temperature and fi are corresponding natural
frequencies.
f1=0.843-0.00039t
f2=1.402-0.00030t
(a) Natural frequencies identified by EFDD versus average temperature
f1=0.843-0.00038t
f2=1.401-0.00029t
(b) Natural frequencies identified by SSI-COV versus average temperature
Figure 4. Natural frequency versus average temperature
According to Fig. 4, it may be concluded that the frequencies identified by both EFDD and
SSI-COV methods can reflect similar global effect of temperature, despite the better
performance of SSI-COV, which allows a better frequency resolution of the estimates.
Although other environmental and operational factors such as wind, humidity and traffic
loading may also affect the natural frequencies, from the viewpoint of long term monitoring,
temperature may be considered the main environmental factor. Statistical analysis of these two
natural frequencies shows that the normal temperature change produces variations of 1.7% and
1.4%, which may mask the change of natural frequency caused by structural damage. To
establish a baseline for long term structural health monitoring, such effect induced by
temperature should be eliminated effectively, and features which are sensitive to damage yet
insensitive to environmental change are necessary.
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4 ELIMINATION OF THE TEMPERATURE EFFECT
4.1
Theory of Principal Component Analysis and Novelty Detection technique
Principle component analysis (PCA) is a multi-variate statistical method. Under the assumption
that the environmental conditions have a linear effect on the identified parameters, PCA
methodology can eliminate such effect. In the current study, PCA analysis is used to remove
temperature effect. Subsequently, novelty detection technique may be used to detect possible
damage. The basic idea of novelty detection is first to build an internal representation of the
structure’s baseline in normal condition, and then examine subsequent data to see if they
significant depart from a normal condition.
Let us consider the matrix Y ∈ R n× N whose column vectors y k are the identified n-order
natural frequencies at time tk,(k=1,2,…..,N, N is the number of samples). A singular value
decomposition of the covariance matrix of Y is
YY T = UΣU T
(3)
Σ1
0
(4)
Σ=
0
Σ2
where U is an orthonormal matrix ( UU T = I ), whose columns define the principal components
and form a subspace spanning the data, Σ is the singular value matrix representing the active
energy of the associated principal components. Matrix Σ can be split in two parts:
Σ1 = diag (σ 12 , σ 22 ...σ m2 ) is a diagonal matrix with the square of the first m singular values on the
diagonal, ranked by decreasing order, and Σ 2 = diag (σ m2 +1 , σ m2 + 2 ...σ n2 ) . Define the indicator:
m
i =1
I=
σ i2
n
i =1
σ i2
(5)
and determine m as the lowest integer such that I > e(%) , where e is a threshold value (i.e.
95%). The meaning of this threshold is as following: m unobserved factors contribute to e% of
the variance in the observed data (Deraemaeker et al 2007).
The first m columns of U are the principal components, which are associated with the most
influencing unobserved factors and constitute a transformation matrix T (loading matrix). T can
project the frequency matrix Y into the environmental-factor characterized space X (scores
matrix)
X=TTY
(6)
The new data can be re-mapped into original space
Yˆ = TX
(7)
The residual error E due to the loss of information while performing the two-way projection
can be calculated as
E = Y − Yˆ
(8)
The new feature vector is given by E, it corresponding to the dynamic features from which
the environmental effects have been removed.
Applying the novelty detection technique to the residual error E, the Novelty Index (NI) can
be defined using the Mahalanobis norm
NI k = E kT R −1 E k
(9)
T
where R=(YY )/N is the covariance matrix of frequency matrix Y.
To detect possible damage, an X-bar control chart (Yan et al 2005; Diego et al 2005) is
constructed by drawing two lines: a centre line (CL) and an additional horizontal line
corresponding to an upper limit (UCL), these are:
C L = NI
(10)
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UCL = NI + ασ
(11)
where NI and σ is the mean value and standard deviation of NI in the reference healthy
state. α is taken as 3, which corresponding to 99.7% confidence.
Two criteria can be employed as damage warning (Deraemaeker et al 2007; Yan et al 2005;
Diego et al 2005): (1) outlier analysis, counting for the percentage of the NI lying outside the
UCL and (2) ratio of NI between healthy and damage state. In the healthy state, the new
vibration features should stay in the hyperplane spanned by the features in reference state.
Percentage of the NI overpassing UCL is rather small and ratio of NI → 1 . On the contrary, with
the emergence of damage, the new vibration features will depart from hyperplane in the
reference healthy state, which will cause a significant increase of outliers and relatively large
ratio of NI .
4.2 Application of Principal Component Analysis and Novelty Detection Methodology
PCA and novelty detection have been already applied in numerical and laboratory models
(Deraemaeker et al 2007; Yan et al 2005; Diego et al 2005). In this paper, this method is used
for the long term monitoring results of Coimbra footbridge.
According to Fig. 4, it can be assumed that temperature has a linear effect on the first two
natural frequencies. Temperature linear effect on frequencies can be also reflected by linear
relation between different frequencies, as shown in Fig. 5. These two natural frequencies
identified by EFDD method constitute matrix Y= [y1, y2]. Substituting Y into Eqs. (3-5), one
obtains the singular values 2.654 and 3.751E-6 with I > 99.9% , which indicates that only one
environmental factor markedly affects the variation of natural frequencies. In the following part,
two sets of frequencies identified by EFDD method both in winter time and in summer time are
investigated to illustrate the potential of removing the temperature effect.
(a) EFDD results
(b) SSI-COV results
Figure 5. Correlation between the first two natural frequencies from 1st, June 2007 to 31st, May 2008
Fig. 6 shows two sets of consecutive samples both in summer and winter. The corresponding
average temperature varies from 18.17 ºC to 34.40 ºC in summer, whereas changes from 0.19 ºC
to 16.46ºC in winter. It is clear that most of the two natural frequencies estimates in winter time
are higher than those in summer time because of temperature effect. PCA and Novelty
Detection are applied to these data and results are shown in Fig. 7. It can be observed that NI in
both two sets stay in the same level, the results of outlier analysis are similar and the ratio
of NI → 1 . The effect of temperature is efficiently removed.
This footbridge is new and no damage is reported from 1st June, 2007 to 31st May, 2008. A
baseline healthy state of this footbridge has been constructed based on the monitoring results
during this period. The successive monitored data from 1st June 2008 to 1st December 2008 is
used to compare with baseline state. The corresponding results of NI are displayed in Fig. 8.
Compared with baseline healthy state, the NI during the period 1st June 2008 to 1st December
2008 still remain small, outlier analysis exhibits similar results and ratio NI are still close to 1,
which means that the footbridge is still in a healthy stage and no damage occurred, as expected.
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winter
summer
winter
summer
(a) 1st frequency
(b) 2nd frequency
Figure 6. Comparison of the first two natural frequencies
both in summer (18.17 ºC -34.40 ºC) and winter (0.19 ºC -16.46ºC)
winter
outlier analysis: 1 %
summer
outlier analysis: 1%
NIw/NIs=0.977
Figure 7. Residual error of data in summer (18.17 ºC -34.40 ºC) and winter (0.19 ºC -16.46ºC)
Set I
Set II
outlier analysis: 0.8 %
outlier analysis: 1.3%
NI I / NI II =0.991
Figure 8. Comparison of residual error of data in
Set I:1st June, 2007-31st May, 2008 and Set II: 1st June, 2008- 1st December, 2008
5 CONCLUSION
This paper mainly discusses the effect of temperature on natural frequencies of a new
footbridge based on long term dynamic monitoring data. Firstly, results from appropriate data
processing confirm that temperature is an important environmental factor that originates a linear
effect on identified natural frequencies. To remove this effect, PCA and Novelty Detection
techniques were introduced. The potential of this methodology is illustrated by two sets of data
in winter and summer time. Finally, long term monitoring data collected during the first year is
used to construct a baseline healthy state. The remaining data is processed and compared with
this baseline state to detect possible damages. As this footbridge is quite new, the ratios
between averaged NI remained close to one, as expected. In the future, possible damage will be
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numerically simulated and the potential of PCA and Novelty Detection techniques will be
further investigated.
ACKNOWLEDGMENTS
The authors acknowledge the financial support provided by the Portuguese Foundation for
Science and Technology (FCT) in terms of Basic Funding of CEC / FEUP, as well as the Ph.D.
Scholarship provided to the first author.
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