Global Positioning System (GPS) with 10-20 Hz sampling rates has become a useful
tool for measuring and monitoring static, quasi-static and dynamic responses
in long-period civil engineering structures exposed to gust-winds, traffic,
earthquakes or temperature variation. Accelerometers have been used extensively
for bridge dynamic monitoring using the force measurements directly. These sensors
are used to sense accelerations. Compared with other surveying systems such
as a surveying total station, accelerometers have some special advantages when
they are used for bridge monitoring. The sampling rate of an accelerometer can
reach several hundred Hz or even higher depending upon application requirements,
which is a very important characteristic when monitoring a bridge with high
dynamics (Meng et al., 2007).
Loves et al. (1995) measured horizontal displacement
history of calgary tower against wind loads using GPS and analyzed the natural
frequency based on such measurements. Celebi (2000)
measured horizontal displacement history of a 44-story building and verified
that the natural frequency of 0-23 Hz as computed from such measurement coincided
with the natural frequency as analyzed from the buildings accelerometer
study. Tamura et al. (2002) demonstrated that
it was possible to directly measure actual displacements using GPS with publicized
accuracy of ±1 cm +1 part per million (ppm) in cases when a buildings
natural frequency is less than 2 Hz and amplitude greater than 2 cm and published
the displacement history of a 108 m steel tower. Breuer
et al. (2002) measured displacement history and natural frequency
of the Stuttgart TV tower against wind loads and suggested the usefulness of
GPS in monitoring safety of high-rise buildings that are thin and long based
on such GPS measurements. But due to the inherent deficiency in the GPS satellite
geometry, multipath, residual tropospheric delay and cycle slips, GPS alone
cannot provide the required positioning precision all the time to meet the requirements
for such a system to detect subtle deformations of structures (Meng
et al., 2007). So Meng et al. (2007)
and Chan et al. (2006) integrated GPS-measured
signals with accelerometer-measured signals to enhance the measurement accuracy
of total (static plus dynamic) displacement response of a structure.
The aims of this study are: (1) to use the wden function available within Matlab library to de-noise the GPS signals; (2) to examine the GPS and accelerometer techniques in deformation monitoring of the bridge towers; (3) to calculate the displacement and frequency of the bridge towers and (4) to evaluate the effects of the applied loads on the bridge tower movements.
BRIDGE DESCRIPTION AND DATA COLLECTION
The Yonghe Bridge links the two cities in China (Tianjin and Hangu). This bridge
was constructed by pre-stressed concrete in December, 1987, closed in October,
2006 because of cracks over mid-span and opened in August, 2007 after rehabilitation.
|| Diagram of Yonghe Bridge and the position of the sensors
|| GPS dynamic monitoring scheme
The Yonghe Bridge has four lanes with the total length of 510.00 m and main
span of the bridge is 260.00 m (Fig. 1). For safety assurance,
a sophisticated long-term Structural Health Monitoring (SHM) system has been
designed and implemented by the Research Center of Structural Health Monitoring
and Control of Harbin Institute of Technology (HIT) to monitor loads and response
of the bridge. The structural health monitoring system for the Yonghe Bridge
comprises a data acquisition and processing system with a total of approximate
179 sensors, including accelerometers, strain gauges, displacement transducers,
anemometers, temperature sensors, weight-in-motion sensors and three GPS's (Fig.
2). The GPS's were permanently installed on the two towers tops of the bridge
and bank near the bridge. A local Bridge Coordinate System (BCS) was chosen
for the analysis and evaluation procedures of the observations performed. In
this coordinate system, the Y-axis shows the traffic direction (span direction),
the X-axis shows the lateral direction and the Z-axis gives the vertical direction
of the bridge. It was assumed that this coordinate system would be beneficial
for the evaluation of performed observations, description of the movement of
the structure and allow a better interpretation of the analysis results as it
is related to the movement directions of the structure.
In this section, the data of South tower of Yonghe Bridge was collected in January 17 2008 from 12:00 to 13:00 p.m. The analysis was based on the data collection in the X and y-directions, since, the movement in these directions are greater than in Z-direction, thus the data in Z-direction were declined.
GPS displacement: For analyzing the signals of GPS measurements, a preprocessing
should be done first. That is to delete noises and extract useful signals. Wavelet
analysis is a strong tool to eliminate noises according to the noise characteristics.
Function wden is exploited to eliminate noises of one dimension time series
in Matlab wavelet analysis packet automatically. There are 4 kinds of method
to select the threshold and soft, hard threshold eliminating noises. Furthermore,
there are global or decomposed layers threshold available for selecting also.
In this study, within the wden function, heursure is used to eliminate and compact
the noises (Yu et al., 2006). Displacement measurements
were performed on January 17, 2008 from 12:00 to 13:00 PM. The original displacement
history measurements in X and Y directions (local coordinates) on the tower
were extracted using wden function as presented in Fig. 3a and
b. Time history of wind and temperature for the bridge is shown in (Fig.
The trend components in the series were investigated from the obtained data
within wden analysis. The trend component in the series represents the long-term
changes related to time and it can be defined by a polynomial function in the
time domain. The transformation of the series without trend components from
time domain to frequency domain is performed using Fast Fourier Transform (FFT)
which is not different from Discrete Fourier Transform (DFT).
||Time history of movement of a southern tower bridge in (a)
X and (b) Y-directions
|| Time history of wind and temperature for the bridge, (a)
speed, (b) direction and (c) temperature
It is an effective and excellent algorithm for the calculation of DFT. Yet,
as DFT has periodic characters, it is assumed that the final sample of the signal
is followed by an initial sample of the signal in the spectrum calculation.
In this case, a spectral leakage occurs as a result of the signal energy leakage
to other frequencies.
In order to minimize this effect, it is proposed to multiply the signal by window function as expressed in Eq. 1. This function characterizes with amplitude slowly approaches zero at the edges before the transformation is performed.
FFT is applied to Y (tip) = Y (ti)* w (i) windowed observations, X (n) FFT coefficients in Eq. 2 are obtained.
For 0≤n≤N - 1, the calculation of X (n) in Eq. 2 requires N complex multiplications and N-1 complex sums. Computing all the N of the X (n) values demands N2 complex multiplications and N2-N complex additions. The FFT coefficients X (n) are in the complex plane but this representation does not aid interpretation. Therefore, the power of the FFT signal is:
where, Pxx (n) values are calculated with the use of Eq.
3 and dominant frequencies in the series are determined by the density frequencies
of the signal. The nature frequency ranged from 0.01 to 0.50 N, where N is the
number of story (Nayeri et al., 2008). In this
case, the tower height of Yonghe bridge is 62.50 m. Approximately, it was assumed
that, the height of story equal to 2.80 m and the tower frequency between 0.22
to 11 Hz. In addition, the frequency components of the bridge towers were calculated
at high of 0.2 Hz. The first mode natural frequencies of the southern tower
bridge were shown in Fig. 5a, b and 7a,
b. It was found that the GPS and accelerometer values in the
X-direction to be 0.32 and 0.41 Hz, respectively, whereas these values were
found to be 0.31 and 1.1 Hz, respectively in Y-direction.
Acceleration measurements: Figure 6a and b
show the values registered by the accelerometer at a 100 Hz-data rate on the
same time of GPS observation. It is possible to note that the output is quite
noisy. In spite of that, the accelerometer values obtained have a absolute range
variation of 2.42 and 0.14 m sec-2 with average 0.01 and 0.00 m sec-2
in X and Y-directions, respectively.
The FFT was applied to a 2x1016 sample of accelerometer data and
the frequency spectrum results are shown in Fig. 7a and b.
Torsional displacement: The tower displacements consist of horizontal displacement of X-axis and Y-axis as well as torsional displacement. For this reason it is assumed that the first coordinates observed denote GPS1 coordinates as (X1, Y1) and next observations are GPSi coordinates as (Xi, Yi), so the torsion displacements T (t) are measured from the first coordinate observed. The T (t) and the coordinates values at time t can be computed as follows:
The computed T (t) values of the tower are shown in Fig. 8a. In general, a tower bridge is assumed a rigid diaphragm with infinite stiffness. Therefore, the distance between two GPS stations (i.e., GPS coordinates) may maintain zero m during the measurement periods. The distance error between two GPS stations during the measurement periods can be computed as follows:
Using this distance error can indirectly evaluate the accuracy of the GPS displacement measurement system. The computed distance error between two GPS stations measurements as shown in Fig. 8b and was maintained with 8 cm range.
|| Fundamental natural GPS signal frequency plot: (a) X-axis
frequency and (b) Y-axis frequency
||Time history of accelerometer the southern tower bridge, (a)
X and (b) Y-directions
||Fundamental natural Accelerometer signal frequency plot, (a)
X-axis frequency and (b) Y-axis frequency
|| (a) Torsion displacement and (b) the distance error between
the first and series positions
Deformation analysis is performed using the certain epoch's results. F-test statistical analysis is adopted to find out the deformation in the tower. In this study, the statistical analysis of the deformations obtained from wden Matlab function is based on the first observation group as shown in Fig. 9 (i.e., this group is considered as a datum group and consists of the first 3 sec observations). The time series of the next groups in X-direction are related to the first group and calculated as follows:
||Test value for the X-direction deformation at the ith estimation
||The difference between the mean of the datum group (c) and the next time
||The variance in the X-direction each 3 sec estimation at the ith
||The variance in the X-direction for the first 3 sec case estimation interval
||The variance in the difference vector
For deformation analysis, the zero and alternative hypotheses are defined as follows:
If Ti ≥ F(1-α, 2, df), F-test depended on the degree of
freedom (df) at the 95% confidence level. It is considered that the difference
vector (di) is significant and that there is indeed a deformation
in the X-direction component (Schroedel, 2002). The same
analysis procedure is also performed in the Y-direction components, as shown
in Fig. 10a and b.
|| Continuous deformation analysis scheme
|| Continuous deformation (blue) statistical tests (a) X-direction
and (b) Y-direction
As shown in Fig. 3, the displacement in X-direction is found to be between -20.02 and 28.91 mm with average value of 1 mm, whereas it was found to be between -13.02 and 23.38 mm with average value of 0.80 mm in Y-direction. Figure 3 also, revealed that the average directional displacement can be construed as a static component of the displacement while its dynamic component can be verified to be approximately 1 mm. The average wind speed during the observation period of the tower displacement measurement using GPS was 2.50 m sec-1 with the prevalent wind direction of 25° (Fig. 4).
These results revealed that the loads and traffic velocities affect the towers movements and the tower was moved in the opposite direction of the wind since the wind speed is equal to 2.50 m sec-1 (normal speed). In addition, it can be seen that the temperature has lower effects on the deformation, since, the difference in temperature was found to be 1.75°C (Fig. 4c).
From Fig. 3, it was found that the residuals between the original and extracted signals are 5.97 cm. This indicated that the noise of GPS signals is high. As well as, it can be seen that the wden function processing caused an increased in the signals accuracy by 20%. Therefore, it's recommended to use wden function in the extraction of GPS signals.
The obtained data from wavelet analysis were used to plot the power spectral
density in X and Y directions as shown Fig. 5 and 7.
From these Fig. 5 and 7 it can be seen that
the power in the X direction is greater than that in the Y direction. This indicates
that the power spectral density is a good parameter to detect the tower movements
and this observation complies with the resulted obtained by Erdogan
and Gulal (2009).
From Fig. 6, it can be seen that the range and average of acceleration in X direction is too high, which indicates that the tower movements in X-direction are critical. In addition, it can be seen that there is a difference between the calculated frequency from GPS and that calculated from Accelerometer signals. This reveals that the GPS system can be used to measure the deformation. Due to its signals errors, GPS can not used to measure the high frequency of dynamic behavior of tower. Thus, accelerometer must be added to the monitoring systems for measuring the high frequency of the structures.
Figure 8 shows that the T (t) is equal to 0.51 radian and the distance errors in almost of the observation period didnt exceed 1cm, which indicates that the tower movements are affected by the traffic loads. In addition, Fig. 8 shows that the GPS signals contain a complex error, which affects the accuracy of the calculated deformation values.
From statistical analysis, it can be seen that the tower movements are very
clear in time periods from 400 to 900 sec in X-direction and 1600 to 2600 sec
in both directions (i.e., X and Y-directions). These indicated that the movements
of tower are not susceptible by the environmental effects and the results cited
in Li et al. (2009) insure this fact.
Based on this limited study, the analysis of the results leads to the following findings:
|| GPS signals noise contains complex errors and the signals
accuracy obtained from the wden function increased by 20%. So, it's recommended
to use wden function in the extraction of GPS signals
||Due to the GPS signals errors, GPS can not used to measure
the high frequency of dynamic behavior of tower. Thus, accelerometer must
be added to the monitoring systems for measuring the high frequency of the
||The power spectral density is a good parameter to detect the
||Based on the statistical analysis, it was found that the traffic
loads are the main factor affects the tower movement, in addition the environmental
effects are not susceptible on it
||GPS can be used as a trustworthy tool for characterizing the
dynamic behavior of the low frequency bridges. With the advance of sample
rate frequency of GPS receivers, the dynamic behavior of the bridges in
high frequency can be measured
This study is financially supported by NSFC (Grant No. 50525823 and 50538020) and MOST (Grant No. 2006BAJ02B05, 2007AA04Z435 and 2006BAJ13B03).