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Bridge Monitoring with Wavelet Principal Component and Spectrum Analysis Based on GPS Measurements: Case Study of the Mansoura Bridge in Egypt Mosbeh R. Kaloop 1; Emad Elbeltagi, M.ASCE 2; and Mohamed T. Elnabwy 3
Abstract: Deformation and expansion joint movement of bridges are among the problems that widely exist in bridge engineering practice. Therefore, it is very important to monitor deformation of bridges and to timely process and analyze the measured data to ensure their safety. Mansoura Bridge, located at El-Mansoura City in Egypt, is selected to demonstrate the utilization of bridge deck deformation monitoring data to analyze and evaluate its safety based on global positioning system (GPS) measurements. The wavelet principal component analysis (WPCA) and spectrum methods are used to establish the time and frequency domain of bridge deformation and analyze the behavior and movement of the bridge under working traffic loads. The following conclusions are obtained in this study: (1) the WPCA can be used to eliminate the GPS observation errors; (2) the three-dimension spectrum gives much richer information for dynamic response; and (3) the deformation and expansion joint movement of Mansoura Bridge are safe under current loads. DOI: 10.1061/(ASCE)CF.1943-5509 .0000559. © 2014 American Society of Civil Engineers. Author keywords: Global positioning system (GPS); Monitoring; Wavelet; Principal component analysis (PCA); Mansoura-Bridge.
Introduction One of the best methods to assess bridge behavior is usually undertaken by means of monitoring. Very often, the measurement of vibration displacements is the simplest way to observe the evolution of a deformation and to analyze the kinematics of the movement, safety state, and the response to the triggering conditions (i.e., traffic). In all cases, measurements have to be made efficiently in terms of time and budget. Bridge deformation refers to the changes the bridge sustains in its shape, dimension, and position. The bridges response to serve loads primarily consists of three components: a static component attributable to mean force, a semistatic component attributable to low-frequency force fluctuations, and a resonant component attributable to force fluctuations near the bridge’s first-mode natural frequency (Li et al. 2006a). Monitoring and detecting such deformations can be determined by using a number of precise methods. Traditional techniques for deformation detection include on-site visual inspection, photogrammetric surveys (terrestrial or aerial), precise conventional surveys, and geotechnical measurements using either continuous data collection or observation epochs. The most recent approach for structural deformation detection utilizes the global positioning system (GPS).
1 Assistant Professor, Public Works Engineering Dept., Faculty of Engineering, Mansoura Univ., Mansoura 35516, Egypt (corresponding author). E-mail:
[email protected] 2 Professor, Structural Engineering Dept., Faculty of Engineering, Mansoura Univ., Mansoura 35516, Egypt. 3 Master Student, Structural Engineering Dept., Faculty of Engineering, Mansoura Univ., Mansoura 35516, Egypt; and Research Assistant, Survey Research Institute, National Water Research Center, Giza 1211, Egypt. Note. This manuscript was submitted on April 3, 2013; approved on December 10, 2013; published online on December 12, 2013. Discussion period open until January 6, 2015; separate discussions must be submitted for individual papers. This paper is part of the Journal of Performance of Constructed Facilities, © ASCE, ISSN 0887-3828/04014071(10)/$25.00.
© ASCE
Noor and Collier (2007) used GPS to monitor the deformation of the West Gate Bridge and also referred to some previous studies that used the GPS for monitoring deflections of structures such as cable stayed bridges (Leach and Hyzak 1994; Larocca 2004), suspension bridges (Ashkenazi et al. 1997; Nakamura 2000; Wong et al. 2001), high rise buildings (Celebi 2000; Brownjohn et al. 2004), and towers (Lovse et al. 1995; Li et al. 2003). Im et al. (2011) summarized the use of GPS technology in structural health monitoring (SHM) for many types of structures. In addition, Kaloop (2010) presented the GPS health monitoring technique for some Chinese bridges. As any other developing technologies, the GPS has its own disadvantages when it is needed to be applied precisely on some engineering applications (Roberts et al. 2002; Gao et al. 2011; Oluropo et al. 2014). A major barrier of applying GPS is the achievable accuracy of their positioning solutions owing to their errors and shaking noises, which is affected by many factors and restraints. Therefore, noise reduction of GPS observations, improvement of the accuracy of the GPS time series, and detection of deformation epochs are the key issues of deformation analysis. This paper focuses on the semistatic displacement analysis of Mansoura Bridge measurements using real-time kinematic (RTK)GPS under working traffic loads. Then, it uses the wavelet analysis to study the safety and vibration state of this bridge in both time and frequency domains based on GPS measurements.
Bridge Description and GPS Measurements Mansoura Bridge is one of three important bridges crossing the Nile River (Damietta branch) in Mansoura city, Egypt, and was constructed in 1991. This bridge connects the cities of Mansoura, Damietta, and Tanta. The total length of the bridge is 2,037.5 m (from point A to B, as shown in Fig. 1) with a 270.0 m span over the Nile River. The total width of the bridge is 21.0 m with two vehicle lanes (8.00 m) for traffic on each direction and two pedestrian walkways (2.00 m) and a middle median of 1.00 m width
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B
(a)
A
(b) Fig. 1. View of Mansoura Bridge and GPS monitoring system: (a) GPS base station; (b) GPS Rover Station (overlay images by Mohamed T. Elnabwy; satellite image from Google Earth, © 2013 ORION-ME, © 2013 Google, Image © 2013 Digital Globe)
[Fig. 2(b)]. The part of the bridge over the Nile, as shown in Fig. 2(a), consists of five spans with a maximum (middle span) span of 80.0 m, two spans of 50.0 m, and another two spans of 45.0 m. These spans are supported by RC piers carried by a pile cap that is based on piles, and the bridge deck is 8.50 m above the Nile River. The bridge is a continuous supported prestressed concrete trapezoidal box-girder as shown in Fig. 2(b). Stiff steel handrails are fixed with the bridge deck as shown in Fig. 2(b). In addition, the bridge deck along with the handrail has been modeled with the finite elements (FEM) and analyzed using ABAQUS software. Fig. 3 shows the effects of the deformation on the handrail along its height. Fig. 3 shows that the change of the deformation on the handrail height is very small and can be neglected and accordingly has no effect on the recorded deformation using the GPS. In addition, a finger plate steel-type expansion joint are used to accommodate the movements of the bridge that result from concrete shrinkage, thermal variation, long-term creep, and that which prevents water and debris infiltration to the substructure elements below [Fig. 2(c)]. An expansion joint device must provide a relatively smooth riding surface over a long service life. Expansion joint devices are highly susceptible to vehicular impact that results as a consequence of their inherent discontinuity. The data presented in this paper were collected using two GPS (rover) receivers clamped at the center of the longest (middle span) span (U2) to study the deformation of deck and at the expansion joint (U1) to study the movements of the deck and the state of the joints of the bridge. These two points are located on the top of the handrail of the sidewalks of the bridge, as shown in Figs. 1(b) and 2(a). The measurement system used is a real time kinematic (RTK) GPS. The base GPS, rover GPS, and radio unit are used to collect raw data at a rate of 1 Hz (Fig. 1). The measuring condition was favorable for the receiver and free of any obstructions at a 15° angle view of the horizon, and at least four satellites were © ASCE
tracked continuously. The time observation for each rover point is approximately 1 h. The GPS base receiver, recording also at 1 Hz, was placed approximately 3.6 km away from the bridge at stable ground, as shown in Fig. 1(a). The data collected were preprocessed using GPS-Trimble software (Trimble 2003). The output of the GPS software was the time series of instantaneous Cartesian coordinates of the rover receiver in the World Geodetic Coordinate System 1984 (WGS84). A local bridge coordinate system was established to be used in the analysis and evaluation of the observed data. The azimuth of the bridge is 5°18′59.76″ (calculated from the data). In this paper, the x-data represents the displacement changes along the longitudinal direction of the bridge, the y-data represents the displacement changes along the transverse direction of the bridge, and the z-data represents the relative displacement change along the altitude direction of the bridge.
Analysis Methods This paper studies the health state of Mansoura Bridge; therefore, two methods are used to analyze the safety of the bridge: the wavelet principal component used to remove the GPS errors and noises, and the time-frequency method used to analyze the behavior state of the bridge based on GPS movement measurement under affected load.
Wavelet Principal Component Denoised Method Principal component analysis (PCA) is among the most popular methods for extracting information from collected data, Also, it is among the most notorious data-analysis tools designed to simplify multidimensional data by tracking new factors supported to
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(a)
(b)
(c) Fig. 2. (a) Elevation; (b) cross section; (c) view of expansion joint and details of Mansoura Bridge (image by Mohamed T. Elnabwy)
capture the main features; and it has been applied in a wide range of disciplines (Bakshi 1998; Ogaja et al. 2003; Aminghafaria et al. 2006). PCA is suitable for movement monitoring, whereas the correlated variables are being measured simultaneously (Ogaja et al. 2003). In addition, the wavelet is a strong tool to eliminate © ASCE
GPS noises according to the noise characteristics (Bakshi 1998; Yu et al. 2006). Also, wavelets have found wide use for signal analysis and noise removal in a variety of fields because of their ability to present deterministic features in terms of a small number of relatively large coefficients (Bakshi 1998).
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Observed GPS signals
Wavelet Transform
killing detail of wavelet from 1 to wavelet level -2
PCA
Similarity measurement
Select eigenvalues greater than 0.05 times the sum of all eigenvalues
Fig. 3. Relationship between deck bridge deformation and handrail height based on bridge FEM
Wavelet Transform
Denoising GPS signal
Reconstruction stage
Training signal presentation Training stage
The PCA transforms an (n × p) data matrix, X, by combining the variables as a linear weighted sum as X ¼ TPT
ð1Þ
where P = principal component loadings; T = principal component scores; and n and p = number of measurements and variables, respectively. In the case of GPS monitoring data analysis, it is assumed that the variables follow a p-dimensional multivariate (recorded coordinates) normal distribution with mean vector μ 0 ¼ ðμ1 ; μ2 ; : : : ; μp Þ and covariance matrix Σ where μi is the mean for the ith variable and Σ is a (p × p) matrix consisting of the variances and covariance of the (p) variables. The wavelets are a family of basic functions that are localized in both frequency and time and may be represented as 1 t−u ψsu ðtÞ ¼ pffiffiffi ψ ð2Þ s s where s and u = dilation and translation parameters, respectively. For the measured data, the wavelet dilation and translation parameters are discretized dynamically and the family of wavelets is represented as 1 t−u ffi ψ pffiffiffiffiffi ffi ψsu ðtÞ ¼ pffiffiffiffiffi ð3Þ 2m 2m where ψðtÞ = mother wavelet. The translation parameter determines the location of the wavelet in the time domain, whereas the dilation parameter determines the location in the frequency domain as well as the scale or extent of the time-frequency localization (Bakshi 1998). The wavelets presented by Eq. (3) may be designed to be orthonormal to each other (Bakshi 1998). Wavelet PCA transforms the data matrix X into a matrix WX, where W is an n × n orthonormal matrix representing the orthonormal wavelet transformation operator containing the filter coefficients (Bakshi 1998; Gumus et al. 2010) W ¼ ½ HL
GL
···
Gm
···
G1 T
ð4Þ
where Gm ¼ 22log n−m × n matrix containing wavelet filter coefficients corresponding to scale m ¼ 1; 2; 3; : : : ; L; HL = matrix of scaling function filter coefficients at the coarsest scale. The matrix WX is of the same size as the original data matrix, but because of the wavelet decomposition, the deterministic component in each variable in X is concentrated in a relatively small number of coefficients in WX, whereas the stochastic component in each © ASCE
Fig. 4. Block diagram of the proposed denoising WPCA system
variable is approximately decorrelated in WX and is spread over all components according to its power spectrum. The relationship between PCA and W X is that the principal component loadings obtained by the PCA of X and WX are identical, whereas the principal component scores of WX are the wavelet transform of the scores of X (Bakshi 1998). The methodology used to eliminate the GPS errors is shown in Fig. 4. It consists of two stages: training and reconstruction stages. In the training stage, the level of the wavelet is transformed using five order symlets to decompose the GPS signals; then the PCA is applied on the GPS signals to construct an autocorrelation matrix for the decomposed signals. The eigenvalues and eigenvectors are calculated from the autocorrelation matrix and then the eigenvalues are arranged in a descending order to select the eigenvectors with eigenvalues greater than 0.05 times the sum of all eigenvalues of signals. Finally, this stage displays the original and reconstructed signals. In the reconstruction stage, the quality of the reconstructed signals (built from the training stage) is checked by calculating the relative mean square errors (which should be close to 100%). From the previous stage, the numbers of retained principal components are presented. These results can improve the signals by removing the noise based on killing the wavelet details at selected levels. The correlations between denoised and original signals were calculated. The correlation returns a value between −1 and 1. If it equals 1, then the signals are perfectly matched. If it equals −1, it thus indicates negative dependency between signals.
Experimental Results In this section, an experimental test is designed. This test contains a Trimble-5700 GPS receiver base and rover (the distance between the rover and the base of the GPS receivers is 10.0 m) over the National Water Research building to reduce the multipath errors. The accuracy of GPS instruments used (Trimble 5700 GPS receiver), which is 1 cm þ1 ppm (length of base line) in the horizontal direction and 2 cm þ1 ppm (length of base line) for the vertical direction, and the sampling frequency rate is 1 Hz (Trimble 2003). The rover GPS is supported on a rotating arm (1.0 m length) moved on an arc track (Fig. 5), the data collected is converted to World Geodetic Coordinate System 1984 (WGS84). The relative coordinate [measured coordinates—mean of coordinates
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Table 1. Comparison of Time History GPS Observation Position Error after Used Filter Method Direction RMS (m) EPP (m) ENP (m) SNR [Decibel(db)] EKF
Fig. 5. Schematic Rover antenna with a rotating arm
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−0.208 −0.166 −0.123 −0.184
33.81 26.93 37.14 27.59
model, the correlation between smoothed and original signals are 0.98 in the x-direction and 0.96 in the y-direction. Also, the quality check of the reconstructed signals are 100% for the xand y-directions, meaning that this model can be used to predict the movement track. As shown in Fig. 6(a), the mean value of errors (observed, smoothed) with approximately 0.3 mm in the x-direction and 0.05 mm in the y-direction are removed. The previous discussion shows that the recorded data has many errors; most of them are multipath errors and receiver noise (HofmannWellenhof et al. 2001; Roberts et al. 2002). Also, the variance of the observed errors are 0.035 and 0.017 mm, whereas the variance of the smoothed are 0.030 and 0.015 mm in the x- and y-directions, respectively. These results show the accuracy of the GPS signals is increased by 16 and 14% in the x- and y-directions, respectively. Also, the correlation between the recorded and smoothed data is high in the two directions. Fig. 5(b) shows that the smoothed and true track of the arm are approximately equal. In addition, the correlation between the smoothed and true track is greater than the observed measurements. Accordingly, the WPCA can be used to denoise and smooth the GPS recorded coordinates for the monitored structure. To precisely analyze and compare results of the simulation experiments, root mean square errors (RMSE), error positive peak (EPP), error negative peak (ENP), and signal-to-noise ratio (SNR) are respectively calculated for WPCA and extended Kalman filter (EKF) methods with Eqs. (5)–(8), and the calculation results are shown in Table 1. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N uX RMSE ¼ t ðxi0 − xi Þ2 =N ð5Þ
0.1
(b)
0.034 0.034 0.029 0.033
i¼1
Original Smoothed True
0.15
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WPCA
x y x y
0.5
EPP ¼ maxðxi0 − xi Þ
ð6Þ
ENP ¼ minðxi0 − xi Þ
ð7Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 i¼1 xi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi SNR ¼ 20 log pP N 0 2 i¼1 ðxi − xi Þ
ð8Þ
where xi , and xi = filtered and measured values of time history GPS observation, respectively; and N = number of observations. Table 1 shows a comparison between the calculated filter of the experimental GPS time history; the maximum SNR and minimum RMSE means a better effect of the denoising method. Accordingly, the WPCA method can be used to filter and smooth the time history of the GPS recorded data
Fig. 6. Smoothed GPS relative coordinates measurement: (a) time series; (b) x and y scatter
Time-Frequency Method of Vibration Signals (DX and DY)] are collected, and the denoised GPS signals based on the WPCA are presented in Fig. 6. In this study, a five-order symlets mother wavelet in WPCA model is used. From the calculated parameters of the simulation © ASCE
The time-frequency analysis is a technique used to study a signal in both time and frequency domains. The GPS observed coordinates are signals along the time that show the movement of structures in three dimensions. The time-frequency analysis of the GPS
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F¼
3 ðn − 2ÞðR1 þ R2 − RT Þ 1þ 8K n − R1 − R2
ð9Þ
where K = concentration parameter; R1 and R2 are resultant of epochs 1 and 2, respectively; and RT = resultant of the combined epochs. The concentration parameter can be obtained from statistical tables using RT (Martin 2007). The calculated F is compared with critical values from the standard F tables. The two mean directions are not significantly different if the measured F-value is lower than the critical F-value (from statistical tables), which depends on the degrees of freedom, and at significance level 5%.
Evolutionary Blackman-Tukey Power Spectrum Results and Discussions The GPS observations of the selected bridge points shown in Fig. 1 are presented in this section. These measurements are taken under working traffic loads (approximately 6,300 vehicles=h) and analyzed based on the methods presented previously. From previous studies (Xiaomin et al. 2008; Moschas and Stiros 2011), the main loads affecting the bridge are traffic loads. In addition,
0.05 Observed Smoothed
X(m)
Evolutionary power spectra have the capability to map changes in the frequency domain. The evolutionary or windowed BlackmanTukey power spectrum is described in Martin (2007) and MATLAB (2008). It computes the spectrum of overlapping segments of the time series. These overlapping segments are relatively short compared with the windowed segments used by the Welch method (MATLAB 2008), which is used to increase the signal-to-noise ratio of power spectra (Martin 2007). Therefore, the windowed Blackman-Tukey method uses the short-time Fourier transform (STFT) instead of the FFT. The output of windowed BlackmanTukey power spectrum is the short-term, time-localized frequency content of the signal. Many researchers used this method to analyze structures movements (Neild et al. 2003; Doukaa and Hadjileontiadisb 2005; Li et al. 2006b; Kaloop 2010; Al-Badour et al. 2011).
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In contrast to the Fourier transform, the wavelet transform uses base wavelets that have smooth ends. Wavelets (as described in the WPCA method) are small groups of waves with a specific frequency that approach zero at both ends (Martin 2007). Because wavelets can be stretched and translated with a flexible resolution in both frequency and time, they can easily map changes in the time-frequency domain (Ogaja et al. 2003; Martin 2007; MATLAB 2008). For more details about principal wavelet and wavelet spectrum, readers are referred to Martin (2007). In this paper, a mother wavelet is used as in Eq. (3), and the number of scales is defined to enable the wavelet transform to be computed. The scales define how much a wavelet is stretched or compressed to map the variability of the time series on different wavelengths (Lovse et al. 1995; Celebi 2000; Ogaja et al. 2003; Martin 2007). Then, the real or complex continuous mother wavelet coefficients are computed using the continuous wavelet transform. Li et al. (2013) and others (Neild et al. 2003; Mohammad et al. 2013) used wavelet timefrequency to analyze the deformation and damage time series of structures.
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Structural analysis is required to determine whether significant movements occurred between the monitoring campaigns. Geometric modeling is used to analyze spatial displacements. Consider two sets of measurements in two epoch measurements x1 and x2 , and it is necessary to compare the two sets of directions and test the hypothesis that these are significantly different. The statistic test of the equality of the two mean directions is the F-statistic (Martin 2007) © ASCE
0
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Wavelet Power Spectrum
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measurements is used to expect the dynamic behavior and the health state of structures. It is traditionally based on the concept of fast Fourier transformation (FFT), which is used to analyze the vibration of mechanical, civil engineering structures, and others (Owen et al. 2001; Neild et al. 2003; Doukaa and Hadjileontiadisb 2005; Martin 2007; Al-Badour et al. 2011; Zhao et al. 2011). In this paper, two time-frequency analysis methods are used: wavelet power spectrum and windowed Blackman-Tukey power spectrum to process 1-Hz GPS vibration measurements. These methods are designed and implemented using MATLAB (2008).
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Fig. 7. WPCA smoothed bridge GPS movement’s measurement: (a) U1; (b) U2
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Wavelet Powerspectrum
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Fig. 9. z-direction 2D spectrum analysis for point U1
Table 2. Statistical Movement of the Bridge
Deck Movements Analysis
U1 U2 Time (min) Fx × 10−5 Fy × 10−5 Fz × 10−10 Fx × 10−5 Fy × 10−5 Fz × 10−10
The receiver coordinates in the three dimensions ðX; Y; ZÞ were transformed into time series of apparent ðx; y; zÞ movements around a relative zero representing the equilibrium level of the monitoring point. Fig. 7 shows the original and smoothed GPS movement observations (long-period) of the two bridge points. The scatter movements in the two directions, x and y, are shown in Fig. 8. From the calculations of the smoothed signals using the WPCA model, the correlation between smoothed and original signals are 0.91 in the x- and y-directions and 0.80 in the z-direction. Also the quality check of the reconstructed signals are 81% for the x- and y-directions and 65% for the z-direction. These results revealed the accuracy of the GPS observations in the three dimensions. Also, it shows that the accuracy in the x- and y-directions are higher than the z-direction. Figs. 7 and 8 show that the static and quasi-static movements of the bridge can be calculated from the GPS observations after smoothing the collected data. In addition, the maximum movements of the bridge in the x- and y-directions are 2.0 and
10–20 20–30 30–40 40–50 50–60
−3.0 −0.9 −1.8 −3.5 −3.7
−0.2 0.4 −0.5 0.06 −0.3
0.2 −1.4 1.1 2.2 2.7
−4.0 −5.4 −3.9 10.9 −5.2
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−1.4 −0.6 −2.9 −1.8 −0.9
Note: Fx , y, z are calculated from Eq. (5) for the three dimensions x, y, and z.
from recorded weather (Tanta station); the temperature change (ΔT) was þ6°C and wind speed was 12 km=h at the time of observation. The thermal expansion of the concrete structure, ΔL, was calculated as 5.76 mm; (ΔL ¼ αLΔT) where ΔL is the length change in meters; α is the linear thermal expansion coefficient and α ¼ 12 × 10−6 ð1=°CÞ for the concrete; and L is the bridge span length in m. © ASCE
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Fig. 10. 3D spectrum GPS movement component of bridge points U1 (X1, Y1, Z1) and U2 (X2, Y2, Z2); see Fig. 9(b) for spectrum key
1.7 cm, and 1.9 and 1.8 cm at points U1 and U2, whereas the mean deformation of the deck at the two points are 0.1 and 0.09 mm, respectively. Also, it shows that the correlation coefficient between the x- and y-directions are low at the two points since the correlation values are 0.44 and 0.36 for points U1 and U2, respectively. The z-direction movement from time zero to 500 s is high; this may be attributed to gross errors. Accordingly, this time period is not considered when calculating the movements at this point. In general, it can be shown that the movements of the monitored points are smaller than the accuracy of the GPS instruments used. Therefore, it can be concluded that the movements of the bridge are safe at the midspan and the expansion joint under current traffic load effects. The statistical analysis of the point movements are calculated based on Eq. (9) as presented in Table 2. In this method, the original GPS deformation data are divided to epochs every 10.0 min, and the statistical analysis of the movements of the bridge are calculated in relationship to the first epoch observation. The F-value calculated in the three dimensions for the two points are less than the critical F-value (3.85) at a level of significance of 0.05 (Table 2). Therefore, the movements of the bridge points are not significant. In addition, the F-value for the z-direction is very small, which indicates that the relative deformation is very small. Accordingly, it is concluded that the deformation and movement of the bridge are very safe.
Deck Low-Frequency Vibration Analysis The displacements graph shown in Fig. 6, at first glance, provides no evidence of any dynamic displacement of low-frequency owing to the long-period noise (Moschas and Stiros 2011). Moschas and Stiros (2013) concluded that a frequency range 0.4 ∼ 5.0 Hz covers a wide variety of dynamic motions, and a frequency range 0 ∼ 0.4 Hz corresponds to semistatic displacement, which can be © ASCE
a result of slow displacements (low frequency) induced by temperature changes and the corresponding semistatic component of displacements. Li et al. (2006a) concluded that the structural response to severe loads mainly consists of three components: a static component due to main forces; a quasi-static component due to lowfrequency force fluctuations; and a resonant component due to force fluctuations near the structure’s first mode natural frequency. The semistatic component is occur on the bridge as a result of the temperature, wind speed and traffic load affecting the bridge for a long period of time (Erdogan and Gulal 2009). So, to extract the full displacement components of the bridge, the GPS observations must be de-noised and used a high sampling rate GPS receiver. In this study, the smoothed signals are used, which calculated from WPCA to extract the semistatic displacement frequency mode (low frequency effect). The two methods: wavelet power spectrum (WP) and Evolutionary Blackman-Tukey power spectrum (EBP) are applied to analyze the vibration of the bridge. Time-frequency for the z-direction deformations at point U1 obtained by the two methods are shown in Fig. 9. As an application of using the WP, the wavelet analysis is used for 120 different scales between one and 120 using a Haar mother wavelet and sampling period of five to convert scales to pseudo-frequencies. For applying the EBP, a window of 70 data points and 50 data points overlap are used with sampling frequency of 1 Hz. The power spectrum density (PSD) for the GPS smoothed signals does not appear when using the WP methods, whereas it is very clear when using the EBP (Fig. 9). Thus means that the sampling frequency is affected on the PSD results in case of the WP. So, the PSD may appear clearly by increasing the sample frequency of the GPS observations. Also, it can be concluded that the EBP is suitable for this study because the low frequency sampling of the GPS used (Fig. 9). Fig. 10 shows the three dimensions spectrum (time-frequency with PSD) of the smoothed GPS movements to analyze the dynamic for the low frequency behavior of the bridge based on the EBP method.
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However, this 3D result gives much richer information. From Figs. 9 and 11 and Table 3, it can be concluded that the first modes of the semistatic displacement component frequency or low deck bridge frequency have the same values (0.02 Hz) approximately. Also, it can be concluded that the bridge in static and semistatic states is safe under current loading conditions. The results also indicate that the 3D spectrum analysis can be applied on the GPS measurements.
x 10
1.2
0.8
0.6
Conclusions
0.4
0.2
0
0.05
0.1
(a)
0.15
0.2
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Frequency (HZ) -7
7
x 10
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Amplitude
1
4
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2
1
0
0.05
0.1
(b)
0.15
Frequency (HZ)
Fig. 11. FFT spectrums of GPS data smoothed: (a) y-direction for U1; (b) z-direction for U2
In this study, wavelet principal component analysis (WPCA) and the spectrum analysis are used to propose and analyze the movements and predict the status of Mansoura Bridge. The 2D and 3D frequencies time-series analysis are used to predict the GPS signals. Based on this study, the analysis of the results leads to the following findings: • The WPCA can be used to denoise and smooth the GPS recorded coordinates in structures monitoring, and it can be used to increase the accuracy of GPS observation by 14–16%. • From the experimental results, an error of approximately 0.3 mm in the x-direction and 0.05 mm in the y-direction are removed; these errors are mainly multipath errors and receiver noise for the used GPS receiver. • The time series indicated that the maximum movements of the deck bridge in the x- and y-directions are 2.0 and 1.7 cm and 1.9 and 1.8 cm at points U1 and U2, and the mean deformation of the deck at the two points are 0.10 and 0.09 mm, respectively. The movement values are within the accuracy of the GPS instrument used, and the movements of the bridge points are not significant at 95% confidence; therefore, the static and semistatic displacement components are safe. • The WP analysis is not suitable for 1-Hz GPS records, whereas the EBP is very sensitive for the same GPS records. The 3D EBP result gave much richer movement information, and it can also be applied on the GPS measurements. • The semistatic displacement component frequency of the bridge deck is approximately 0.02 Hz and corresponds to lowfrequency force fluctuations.
Acknowledgments Table 3. Semistatic Displacement Component Frequency Mode of the Bridge Deck U1 Monitoring points Frequency (Hz)
U2
x
y
z
x
y
z
0.021
0.021
0.022
0.022
0.021
0.021
The 3D framework reveals not only the timing but also the relative strength of the dominant frequencies. It, also, gives the temporal evolution of the full frequency spectrum (Li et al. 2006b). Fig. 10 is a 3D analysis applied to the GPS observation of the three coordinates for the two points, showing the high PSD-induced frequency response, each with a certain bandwidth. Also, the high PSD is not repeated on the other frequency plane. In addition, high PSD is continuous with time observation and the PSD values are the same values approximately for the three directions at the bridge monitored points. The upper line tracks the dominant frequencies. Ignoring all the other frequencies and projecting this line onto the time-frequency plane gives the same result as Figs. 9 and 11. © ASCE
This research was supported by Mansoura University, Egypt.
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