The construction of prestressed concrete bridges has witnessed a steep increase for the past 50 years worldwide. The constructed bridges exposed to various environmental conditions deteriorate all along their service life. One such degradation is corrosion, which can cause significant damage if it occurs on the main structural components, such as prestressing tendons. In this study, a novel non-destructive evaluation method to incorporate a movable yoke system with denoising algorithm based on kernel principal component analysis is developed and applied to identify the loss of cross-sectional area in corroded external prestressing tendons. The proposed method using denoised output voltage signals obtained from the measuring device appears to be a reliable and precise monitoring system to detect corrosion with less than 3% sectional loss.
Keywords: denoising; kernel principal component analysis; corrosion; external tendons; prestressed concrete bridges
Prestressed concrete (PSC) bridges have gained increasing popularity and are widely erected all around the world owing to numerous features superior to traditional reinforced concrete bridges, for example, their economical competitiveness, the possibility to achieve longer, lighter, and slenderer structures, and to control cracks more effectively, and so on. The PSC bridge introduces compressive forces in concrete before loads are applied to counteract or reduce the tensile stresses imposed in the structure during service. Since the strength of concrete in tension is much smaller than that in compression, the PSC bridge is known to be very effective, especially in terms of design and construction. The compressive preloading under construction is realized by a procedure called prestressing that tensions high-strength steel tendons installed in the tension regions of concrete [[
In Korea, the first PSC bridge was constructed in the early 1960s and, since then, has become the dominant type of bridge erected in the country [[
In 2016, several PSC bridges along the Inner Circular Highway in Seoul, Korea, were reported to have suffered a rupture of their external tendons. The urgent measure was taken to investigate the cause of damage and find solutions. This accident triggered numerous Korean engineers and researchers to recognize the importance of maintenance as well as the need for reliable monitoring methods detecting corrosion of PSC bridge tendons.
Various non-destructive evaluation (NDE) methods have been developed to detect corrosion in the PSC bridge tendons. The acoustic emission (AE) technique utilizes AE sensors to detect released energy when the tendons are fractured under external loads [[
The guided ultrasonic wave is applied to the inspection of corrosion in pipelines and bridge tendons. This method uses vibrational characteristics of the ultrasonic wave and has been widely adopted to detect corrosion in pipelines [[
Magnetic methods utilize electromagnet-based sensors to detect the change in the magnetic properties caused by corrosion. This magnetic method is proved to be more effective than other methods, such as line scanning thermography, ground-penetrating radar, impact echo, and ultrasonic shear-wave testing [[
The MFL method is a magnetostatic measurement technique developed by Kusenberger and Barton [[
Accordingly, this study adopts the MFL method to develop a novel NDE method to identify corrosion in response to the aforementioned necessity. The target elements are external tendons, i.e., tendons that are not buried inside the concrete structure but exposed to the external environment. Each tendon is composed of a total of fifteen seven-wire strands consisting of 105 wires, i.e., 15 × 7 = 105, and the corrosion is assumed to cause loss of cross-sectional area of the tendon.
The developed device to detect the leakage caused by corrosion is a non-contact movable yoke system. The system consists of a yoke-shaped electromagnet with hinges and two coils. This electromagnet is designed to introduce a magnetic field or magnetic flux in the external tendons between its arms. The sectional loss is detected from the variation of the measured magnetic flux at the location of corrosion. The MFL signals are acquired from a search coil (B-coil) and a Chattock coil (H-coil) with a data acquisition (DAQ) system.
The signals obtained from the yoke system are inevitably contaminated by noise during the monitoring processes, such as data measurement, acquisition, transmission, etc. The noise often cloaks valuable information inherent in the signals and may lead to the erroneous prediction of damage states for the structure being monitored. Therefore, eliminating the noise is essential for constructing a robust monitoring system.
Denoising aims to remove noises from polluted measurements while retaining the significant information in the signals. The simplest denoising method is averaging. The averaging method, as the name implies, just adds all the signals available and computes mean values. Due to this simplicity, the averaging method is widely used to get rid of the noise included in the measured signals.
The main drawbacks of the averaging method, however, are: (
Filtering methods using low-pass filters, high-pass filters, band-pass filters, etc., are also excluded in this study because the moving velocity of the measuring device is not maintained constant due to manual operation during the test. Instead, digital image denoising methods are investigated because the methods employ pixel values as input vectors similar to MFL signal vectors, and the input data are irrespective of constant time steps.
Numerous denoising methods have been developed to improve digital images and signals [[
Kernel principal component analysis (Kernel PCA) is a nonlinear generalization of linear principal component analysis (LPCA). The denoising capacity of the Kernel PCA is known to be better than that of the LPCA [[
This paper is organized as follows. Section 2 explains the test setup to obtain signals. Section 3 describes the mathematical derivation of the proposed Kernel PCA denoising algorithm. Section 4 presents the monitoring results of the sectional loss of the tendons, and the conclusion is provided in Section 5.
When a yoke-shaped electromagnet is connected with an external tendon, as shown in Figure 1a, a magnetic field is induced in the external tendon acting as a magnetic body. A magnetic circuit of the electromagnet and the tendon is also formed with magnetic flux. The corresponding magnetic flux is the magnetic flux density times the cross-sectional area of the magnetized external tendon.
At damaged locations of the external tendons, however, magnetic field leakage occurs and decreases the magnetic flux from that obtained from pristine tendons. The variation of the magnetic flux produces an electromotive force or induced voltage in the exposed conductor. Therefore, a moving electromagnet and sensing coils (conductor) along the external tendon can be used to detect the variation of the magnetic flux or loss at the section.
The test system consists of a yoke-shaped electromagnet with hinges, two sensing coils, and an external tendon made of fifteen seven-wire strands (see Figure 1a). The electromagnet is powered by a core made of laminated silicon steel plates with 900 turns of the magnetizing coil at the center. At the end of each pole of the yoke, a half-cylinder steel shell cover is connected with a hinge for a detachable connection to the external tendons. This electromagnet can generate a magnetic field in the external tendons between its arms.
For MFL signal detection, two different sensing coils are employed. The first one is a typical search coil (B-coil) with 10 turns to measure the magnetic flux variation along the external tendon, and the second is a Chattock coil (H-coil) with 3225 coil turns to measure the magnetic potential. The schematic of the DAQ system used to process the signal is shown in Figure 1b.
Typical tendons made of fifteen seven-wire strands enclosed in a high-density polyethylene (HDPE) duct are considered in the test, as shown in Figure 1a. Damage of the tendon is simulated by cutting off a length of 56 mm from three or six wires out of the 105 wires to create 2.86% and 5.71% reduced sections, respectively (Refer to Table 1). Concrete grouting is not applied inside the HDPE duct because the magnetic behavior of concrete is assumed to be the same as ambient air because both materials have practically the same specific magnetic permeability.
Three independent tests are performed for each external tendon with a reduced section. In each test, direct current (DC) of 6 ampere (A) is applied to the magnetizing coil, and the signals from both B-coil and H-coil are measured. The yoke is moved at a speed of about 70 mm/s on average.
For signal-based damage detection, proper denoising methods should be employed to eliminate the noise that is inevitably included in the measured signals. In this study, Kernel PCA is adopted for the purpose of denoising the measured noisy signals. For successful denoising by Kernel PCA, the transformation of one-dimensional signals to multidimensional ones and the pre-image estimation are required. This section describes the mathematical backgrounds of Kernel PCA, embedding transformation, and pre-image findings.
Principal component analysis (PCA), also known as the Karhunen–Loève transformation, is a widely used mathematical method for dimensionality reduction. PCA transforms the original data onto new linear or nonlinear orthogonal coordinates that can maximize data variance, and dimensionality reduction is achieved by extracting and employing only the first few meaningful principal components from transformed variables [[
The first principal components can be computed by projecting the transformed data onto the eigenvector corresponding to the largest eigenvalue of the data covariance matrix, and the succeeding principal components are calculated similarly by projecting the transformed data onto the subsequent eigenvectors. After calculating and sorting the entire eigenvalues in descending order, only a part of the principal components associated with the largest eigenvalues is extracted on the basis of the contribution, and the rest is pruned. During this procedure, dimensionality reduction, as well as denoising, are achieved because the contribution of noise represented by the smallest eigenvalues is considered a minimum [[
Linear principal component analysis (LPCA) projects the transformed data onto new linear coordinates, while nonlinear principal component analysis (NLPCA) adopts nonlinear coordinates. Since NLPCA searches for nonlinear latent correlation, which may be seemingly indistinguishable but immanent in the original data, the performance of NLPCA has been reported better than that of LPCA [[
In this study, Kernel PCA is selected and employed among many types of NLPCA, such as self-organizing mapping [[
Let x
In LPCA, data covariance matrix C
(
Hereafter, bold italic and capital letters, e.g., x
In Kernel PCA, nonlinearity is achieved by using a nonlinear mapping function, ϕ (⋅). The transformed data, i.e., ϕ (x
(
Then, all the remaining derivations are similar to that of LPCA. The maximization of the data covariance matrix consisting of transformed data turns into solving an eigenvalue problem:
(
where w and λ are eigenvectors and eigenvalues, respectively.
Since the eigenvectors w in the transformed space with non-zero λ lie in the span of ϕ (x
(
After substituting Equation (
(
where
From this equation, N number of eigenvalues and corresponding eigenvectors, such as λ
(
The kernel function, k (x
A dot product, ϕ (x
Calculating a kernel function instead of a dot product is computationally cheaper because it can be conducted in a lower-dimensional space. As can be seen in the procedure, the linear method, e.g., LPCA, is simply extended to a nonlinear one, e.g., Kernel PCA, even without explicit construction of the nonlinear mapping in high-dimensional space. This progress enables us to formulate a nonlinear algorithm in an easier way by specifying and substituting a proper kernel for a dot product [[
For the nonzero eigenvalues of positive-semidefinite kernel matrix, K, the k
(
where (α
So far, the measured data are assumed to satisfy a centering condition, i.e., Σ
(
The k
When the noisy signals of x are acquired, the aforementioned Kernel PCA maps the signals to ϕ (x), discards components of little contribution, and estimates the kernel principal components and corresponding eigenvectors w using only the remaining part of eigenvalues. Through this restrictive selection process to adopt only a small portion of components, the major structure in the measured dataset is captured in the chosen kernel principal components, and the noise associated with pruned eigenvalues can be removed in an effective way. The detailed selection method is described in 4.2.
The denoised signals in the transformed space can be expressed by the projections onto the selected eigenvectors w
(
where ϕ
Then, the denoised signals named pre-image can be calculated by the inverse mapping of the projection in Equation (
(
where
Note that z also minimizes the distance between ϕ
(
The extremum satisfying Equation (
(
For the Gaussian kernel employed in this study, Equation (
(
where
The denominator in Equation (
The aforementioned Kernel PCA is hardly applied to one-dimensional signals, i.e., m = 1 for x
Let d(t) ∈ R
(
where m (<N) is an embedding dimension. These lagged vectors x
(
Once the noise is eliminated from the lagged vectors x
To resolve the discrepancies, diagonal averaging, i.e., replacing each element in the diagonal with the average of the corresponding diagonal terms, is implemented, resulting in another Toeplitz matrix T
The detailed procedure to detect corrosion is summarized in this subsection. First of all, the non-contact movable yoke system collects one-dimensional MFL signals d(t) from bridge tendons, and multidimensional lagged vectors x
The proposed Kernel PCA denoising method is applied to experimental datasets to investigate its capability to detect an abnormality, which is simulated by the reduction of cross-sectional area in PSC bridge tendons. The sectional loss in this study is assumed to be caused by the corrosion of wires inside the bridge tendons.
The data are obtained from the PSC bridge tendons having two different sectional loss levels, i.e., 2.86% and 5.71% loss indicated to as TSL1 and TSL2 cases, respectively. The loss is simulated by cutting three and six wires at around 1300 mm distance from the left end among the 105 wires enclosed in a tendon duct. For each case, three consecutive tests, e.g., TSL1-1, TSL1-2, and TSL1-3, are carried out to collect the output voltage from B-coil and H-coil.
The movable yoke system is mounted on a small cart that is moved manually along the bridge tendon to measure the output voltage. Note that the operating speed for each test and location of each data point do not perfectly coincide with each other, even though great attention has been paid to take measurements under as identical conditions as possible to increase the accuracy. For example, the number of data points obtained for the exploited device to move 1540 mm is 281, 400, and 341 for TSL1-1, TSL1-2, and TSL1-3 cases, respectively. The location of each data point is computed using the average moving distance between adjacent points, e.g., 1540/(N − 1), where N is the total number of data points.
First of all, a preliminary data analysis using noisy measured signals of TSL2-1 containing larger sectional loss is conducted to confirm the necessity for denoising (Figure 4). The noisy output voltage obtained from B-coil shows unexpected high and sharp signals, as indicated by the two red arrows in Figure 4a. Note that the abnormality, regardless of damage, is observed for both cases having 2.86% and 5.71% sectional losses. The first unsuspected high voltage from B-coil seems to be caused by a sudden movement of the cart, while the occurrence of the second peak is quite strange and unaccountable. Furthermore, the first peak value of H-coil voltage near the first arrow becomes negative, i.e., opposite to that of B-coil, which does not accord closely with the fundamental theory. In other words, the output voltage for both coils tends to move up and down near damaged sections, and the measured values approach zero otherwise [[
(
where Re [·], b (t), and h (t) denote an operator to extract only the real part, output voltage from B-coil and H-coil, respectively. The defined damage index must be very effective to (
Figure 4a plots the B-coil and H-coil signals obtained from the TSL2-1 case with the largest, i.e., 5.71%, sectional loss. Then, the damage index is applied to those noisy signals in order to identify the damage, as shown at the bottom of Figure 4a. Professional engineers might confirm the location of damage from the second rectangular window indicated by a dashed line (near the white arrow) in Figure 4a, while false prediction could also be concluded from the first as well as second dashed windows. Note that the TSL2-1 dataset with larger sectional loss is employed in order to emphasize the necessity of denoising. If the severity of sectional loss becomes smaller, for example, 2.86% for TSL1-1, TSL1-2, and TSL1-3, the distinction between true and false identification of damage may get arduous.
Figure 4b describes the averaging results of the entire three TSL2 signals and shows that the included noise is not cleared by the averaging method. The poor denoising performance results from the fact that the measuring locations are not identical for each test due to different moving speeds of the yoke system, and the number of repeated tests is insufficient. To overcome the restriction, the Kernel PCA denoising method is proposed.
To apply the Kernel PCA denoising to MFL data, three parameters should be determined in advance. Those parameters are:
-
(1) The number of eigenvalues, i.e., the extracted number of kernel principal components, _I_k_i_ for KPC_I__SB_k_sb__i_ (_B__I_x_i_
i ), - (
2 ) a Gaussian width of the chosen kernel, σ2 for - (
3 ) an embedding dimension, m.
First of all, the number of kernel principal components is decided on the basis of the average eigenvalue approach [[
The detailed procedure and results of the approach are demonstrated using the TSL 1-1 dataset. Figure 5 shows an average eigenvalue indicated by a red dashed line and the ten largest eigenvalues extracted by Kernel PCA from TSL1-1 B-coil measurements with the percentage contribution of the first five eigenvalues, i.e., 20.9%, 16.6%, 5.8%, 4.8%, and 3.5%, respectively. Note that the first five eigenvalues already correspond to 51.7%. Only 47 eigenvalues out of 281 non-zero eigenvalues are adopted for denoising, which is equivalent to 16.7%. The assessment procedures are to:
- (
1 ) Compute an average of entire normalized eigenvalues, which are arranged in descending order of eigenvalue size, - (
2 ) extract and add eigenvalues of which the magnitude is larger than the average, - (
3 ) calculate the percentage contribution of the chosen eigenvalues.
This selection process is repeated for each value of σ
As a result, optimal values of two parameters, such as the number of eigenvalues and the Gaussian width σ
The embedding dimension m can also be estimated considering mutual interaction with the number of eigenvalues and the Gaussian width. For a specific value of the embedding dimension, the Kernel PCA denoising algorithm searches for an optimal combination of the number of eigenvalues and the Gaussian width, which results in contribution exceeding 80% and its difference below 1%. When the embedding dimension m is 5 instead of 10, the best estimation for k and σ
The proposed Kernel PCA denoising algorithm with the pre-defined damage index is applied to MFL measurements to detect the sectional loss in the prestressing tendons. When the obtained measurements are provided, the Kernel PCA algorithm starts to seek the best combination of the number of kernel principal components and the Gaussian width, which is determined on the basis of two criteria: (
For sectional loss monitoring, six datasets are prepared in this study. TSL1 and TSL2 series datasets are measured from bridge tendons having 2.86% and 5.71% sectional loss, respectively. The number of data points for TSL1-1, TSL1-2, TSL1-3, TSL2-1, TSL2-2, and TSL2-3 is 281, 400, 341, 331, 331, and 311, respectively. Different numbers of input data for each test are caused by dissimilar moving velocities of the movable yoke system carried by hands.
First of all, TSL1-1 data with a 2.86% loss are used to verify the denoising capacity. Note that measurements with smaller cross-sectional loss are employed to investigate the denoising performance. Figure 8 represents TSL1-1 noisy signals, denoised signals, and eliminated noise from (a) B-coil and (b) H-coil output voltage, respectively. As shown in the figure, the proposed Kernel PCA algorithm can effectively separate inherent noise from the original signals and provide reliable denoised signals, even when less than 3% sectional loss is involved.
Figure 9 demonstrates TSL1-1 and TSL2-1 signals with a sectional loss of 2.86% and 5.71%, respectively. Measured B-coil and H-coil signals, denoised signals, and damage index for TSL1-1 and TSL2-1 are presented at the top, middle, and bottom, respectively. The damage index shown at the bottom for both TSL1-1 and TSL 2-1 addresses that (
The diagnosed results for the entire six cases are described in Figure 10. For each case, the identified corrosion locations are depicted by the trough between two adjacent peaks and marked by red arrows. The average detected location for TSL1-1, TSL1-2, and TSL 1-3 signals is estimated to be 1293 mm, i.e., an average of 1296 mm, 1228 mm, and 1354 mm. For other tests of TSL2, the mean location is assessed to be at 1289 mm from the left, i.e., mean of 1312 mm, 1281 mm, and 1274 mm. Note that a good estimation is achieved by the proposed method using measured signals with heavy noise.
Based on the results obtained so far, it can be concluded that the proposed NDE method incorporated with the non-contact movable yoke system, the Kernel PCA denoising, and damage index is capable of identifying corrosion and detecting its location, even when the severity of sectional loss caused by corrosion is less than 3%. Although the present results are satisfactory enough, succeeding research is underway to develop lighter sensing systems that move at a constant speed without sacrificing monitoring accuracy.
This paper presents a novel method to detect corrosion in external prestressing tendons. The proposed method utilizes measured output voltages from B-coil and H-coil of a movable yoke system and eliminates immanent noise from the measurements by the Kernel PCA denoising algorithm. Then, a trough between adjacent twin peaks represented by a damage index using denoised signals indicates the occurrence of corrosion and its location. Through this whole process, it is demonstrated that the proposed method is able to identify and locate loss of cross-sectional area caused by corrosion, irrespective of (
The main procedure of the proposed method can be summarized as follows: (
The proposed Kernel PCA denoising method incorporated with the damage index is applied to the measured noisy output signals from B-coil and H-coil. When the noisy output voltage is utilized, a larger magnitude of the sectional loss, e.g., 5.71%, is hardly identified, or false prediction may be observed. One of the famous denoising methods, i.e., averaging, is shown to produce bad denoising results, either. On the other hand, the proposed methodology is able to provide reliable and precise information on the occurrence and the location of corrosion, even when the severity of sectional loss caused by corrosion is under 3%, e.g., 2.86%, as well as 5.71%.
On the basis of the monitoring results achieved in this study, the proposed Kernel PCA denoising method incorporated with the suggested damage index has demonstrated its ability to identify and locate loss of cross-sectional area incurred by corrosion. The developed novel non-destructive evaluation technique can be readily applied to corrosion detection occurring at prestressing tendons of the prestressed concrete bridges. Additional research is underway to apply the proposed method to in-service prestressed concrete bridges.
Graph: Figure 1 Non-contact movable yoke system: (a) Magnetic field leakage signal test setup with a damaged section; (b) Data acquisition system.
Graph: sensors-20-05984-g001b.tif
Graph: Figure 2 Denoising process used in this study: (a) Data in original space; (b) Data in transformed space.
Graph: Figure 3 A corrosion detection process proposed in this study.
Graph: Figure 4 Sectional loss monitoring using noisy signals TSL2-1 from B-coil and H-coil: (a) Measured noisy signals (top) and damage index before denoising (bottom); (b) B-coil and H-coil signals after averaging.
Graph: Figure 5 Ten largest eigenvalues with percentage contribution of the first five eigenvalues and average of entire eigenvalues extracted by Kernel PCA. PCA, principal component analysis.
Graph: Figure 6 Contribution of above-average eigenvalues for each Gaussian width (top) and the change of contribution between the adjacent sum of chosen eigenvalues (bottom).
Graph: Figure 7 Denoised signal (top) and damage index (bottom) when: (a) m = 10; (b) m = 5.
Graph: Figure 8 Measured signals (top), denoised signals (middle), and separated noise (bottom) from TSL1-1: (a) B-coil; (b) H-coil measurements.
Graph: Figure 9 Measured (top) and denoised signals (middle) with computed damage index (bottom) for: (a) TSL1-1 with 2.86% sectional loss; (b) TSL2-1 with 5.71% sectional loss.
Graph: Figure 10 Damage index using denoised signals with: (a) 2.86% sectional loss; (b) 5.71% sectional loss.
Table 1 Reduced section of the external tendon.
Tests Damage DC (A) Remark Section Loss (%) Cutting Length (mm) TSL1-1 2.86 (3 wires) 56 6 Trial 1 TSL1-2 Trial 2 TSL1-3 Trial 3 TSL2-1 5.71 (6 wires) 56 6 Trial 1 TSL2-2 Trial 2 TSL2-3 Trial 3
Table 2 Estimated parameters when m = 10 (No. of KPCs represents the number of kernel principal components).
Tests B-Coil H-Coil No. of KPCs No. of KPCs TSL1-1 47 90 27 60 TSL1-2 66 90 72 60 TSL1-3 45 90 36 40 TSL2-1 76 180 69 160 TSL2-2 42 110 33 70 TSL2-3 69 110 50 100
Table 3 Estimated parameters when m = 10 and m = 5.
Tests B-Coil H-Coil No. of KPCs δ2 No. of KPCs δ2 TSL1-1 ( 47 90 27 60 TSL1-1 ( 38 40 27 20
C.K.O. developed the system and the algorithm, wrote the manuscript; C.J. led the experiments, fabricated and set up experimental devices, supervised research, and revised papers; J.W.L. operated experimental devices and measured data; K.-Y.P. participated in the experiments, pre-processed measured data, and revised the paper. All authors have read and agreed to the published version of the manuscript.
This research was funded by the Korea Institute of Civil Engineering and Building Technology (KICT) of Korea, Project No. 2020-0033.
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
output voltage from B-coil data covariance matrices one-dimensional measured MFL signal at time one-dimensional denoised MFL signal at time output voltage from H-coil kernel matrix modified kernel matrix KPC extracted number of kernel principal components kernel function embedding dimension number of data points Toeplitz matrix final Toeplitz matrix after diagonal averaging Toeplitz matrix before diagonal averaging lagged vector constructed from measured MFL signals, denoised lagged vector after diagonal averaging pre-image , eigenvectors λ eigenvalues Gaussian width nonlinear mapping function image of
By Chang Kook Oh; Changbin Joh; Jung Woo Lee and Kwang-Yeun Park
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