A Method for the Damage Detection of Pile Foundation in High-Pile Wharf Based on A Curvature Mode Deletion Model

As the top of the pile foundation in high-pile wharf is connected to the superstructure and most of the pile bodies are located below the water surface, traditional damage detection methods are greatly limited in their application to pile foundation in service. In the present study, a new method for pile foundation damage detection is developed based on the curve shape of the curvature mode difference (CMD) before and after damage. In the method, the influence at each node on the overall CMD curve shape is analyzed through a data deletion model, statistical characteristic indexes are established to reflect the difference between damaged and undamaged units, and structural damage is accurately detected. The effectiveness and robustness of the method are verified by a finite element model (FEM) of high-pile wharf under different damage conditions and different intensities of Gaussian white noise. The applicability of the method is then experimentally validated by a physical model of high-pile wharf. Both the FEM and the experimental results show that the method is capable of detecting pile foundation damage in noisy curvature mode and has strong application potential.


Introduction
Most of the piled piers in service were built in the last century. After decades of use under harsh operating environments and diverse loads, many high-pile wharf pile foundations have been damaged, with some severe cases (Wang and He, 2017). If such damage is not timely detected and repaired, catastrophic casualties and property losses may occur. At present, methods for the damage detection of pile foundation mainly include high-strain, low-strain and local methods (ultrasonic methods, electromagnetic radar methods, etc.) (Li et al., 2018;Ding et al., 2011;Liu et al., 2012). However, as the top of the pile foundation in highpile wharf is connected to the superstructure and most of the pile bodies are located below the water surface, the above methods cannot realize the non-destructive and rapid detection of pile foundation damage.
Damage detection based on dynamic signatures is considered to be the most promising method for non-destruct-ive testing of structures, integrating structural vibration theory, vibration testing technology and data processing technology (Jahangiri et al., 2019;Ren et al, 2006). Many studies have applied dynamic signatures damage detection to the health testing and monitoring of high-pile wharf (Sun et al., 2011;Boroschek et al., 2011). In dynamic signatures, the curvature mode shows high sensitivity to local damage and has been the focus of structural engineering research in the last few years. Pandey et al. (1991) was the first to use the curvature mode to detect the damage of cantilever beams and simply supported beams. Since then, researchers have conducted damage detection research with different forms of beam, arch and plate by curvature mode (Cao et al., 2014;Janeliukstis et al., 2017). The curvature mode detection method has also been gradually expanded from the original one-dimensional structure damage detection to multidimensional structure damage detection (Xu et al., 2019), and gradually applied to railway track, bridge, ocean riser and other practical engineering fields (Zhao et al., 2017;. Several researchers have aimed at overcoming the disadvantages of the curvature mode damage detection method, such as poor robustness and the need for baseline data (Xu et al., 2019). Ahmad et al. (2019) used curvature mode analysis combined with filtering technology based on the gap smoothing method to detect multiple damage on a plate structure. Ramesh and Rao (2019) studied the square curvature mode shape of a beam structure and proposed a damage detection method without the baseline data. Furthermore, Cao et al. (2014) established a robust damage detection method based on the curvature mode with wavelet analysis and Teager energy theory. However, these improvements are mostly in the theoretical stage and have not been applied to engineering practice.
To address the curvature mode difference (CMD) limitations of high sensitivity, poor robustness and unrealized automatic recognition in practical engineering detection applications, firstly, we formulate a new damage detection method that does not directly assess damage through the CMD value. Based on the curves shape of CMD, the method fully considers the local perturbation effect of damaged node (Ciambella and Vestroni, 2015), analyzes the influence at each node on the overall CMD curve shape through data deletion model, and uses relative influence intensity as the damage detection index. Secondly, we designed a finite element model (FEM) with various damage types to pile foundation in high-pile wharf and applied the method to it. The efficacy of the method was verified by the FEM, and the robustness of the method was verified by adding noise on the basis of the FEM. Finally, we applied the new method to a physical model of damaged high-pile wharf to verify its practicability.

Characteristics of the CMD curve
Several studies (Nie, 2012;Dessi and Camerlengo, 2015) have shown that in the case of structural damage, the CMD of the damaged region shows an inverted 'V' shape; i.e., the curvature mode at the damaged unit is significantly enhanced, whereas that at the adjacent unit is extensively suppressed, although the degree of suppression is much smaller than the degree of enhancement at the damaged unit. In the study of Nie (2012), a model of 320 units of dome structure was established, and the damage degrees of the 83th unit were set as 10%, 20%, 30% and 40%. The first-order CMDs in all cases are shown in Fig. 1. The inverted 'V' shape appeared at the damage unit, and the decrease in adjacent units accounted for approximately 5%−20% of the increase in the damaged unit under different damage conditions. The existing methods of curvature mode damage detection only determine the damage unit from the 'peak' of CMD curve increase at the damage location and do not con-sider the small changes in the adjacent units. This study establishes a new damage detection method that evaluates the morphological characteristics of CMD curves. The curvature modes that increase in the damaged unit and decrease in adjacent units are considered comprehensively in the method to reflect the difference between the damaged and undamaged units.

Inverted 'V' shape recognition based on the data deletion model
The so-called deletion model is a method to analyze the differences between the model that deletes the data and the original model. This method can effectively detect the inverted 'V' shape in the CMD curve. Take the following two data sets as examples to illustrate the detection effect of the data deletion model.
(1) As shown in Table 1 and Fig. 2, a 'peak' appears at both Points 7 and 16, but the difference is that, an inverted 'V' shape appears at Point 7 because of the decrease in the adjacent point, whereas no inverted 'V' appears at Point 16.
(2) As the damage size and the density of measuring points are different in the actual detection, the inverted 'V' shape may appear through the combination of continuous multi-points. As shown in Table 2  Based on the data in Table 1, the regression curve was established by using cubic polynomial. The regression curve of the original data, the data after deleting only Point 7 and  the data after deleting only Point 16 are shown in Fig. 4. As can be seen from Fig. 4, there is a significant difference in the change degree between the regression curve without Point 7 and that without Point 16, i.e., even if the two points have the same values, they have different influence on the regression curve. To quantify the change of the curves, the change of the regression coefficients is defined as follows: where is the original data regression coefficient vector, is the regression coefficient vector after a data deletion. after deleting Point 7 and after deleting Point 16 are 0.0714 and 0.0426, respectively. Obviously, the change after deleting Point 7 is 1.6761 times the change after deleting Point 16. Based on the data in Table 2, cubic polynomial was still used to build a regression curve. The regression curve of the ∆C original data, the data after deleting only Point 7 and the data after deleting only Point 16 are shown in Fig. 5. As can be seen from Fig. 5, there is still a big difference in the change degree of the regression curve, after deleting Point 7 and after deleting Point 16 are 0.0611 and 0.0445, respectively. Obviously, the change after deleting Point 7 is 1.3730 times the change after deleting Point 16. Through the above two examples, it can be seen that the point with the inverted 'V' shape has a greater influence on the regression curve. The data deletion model can effectively identify the influence of data points, so as to realize the inverted 'V' shape detection in the CMD curve. Therefore, a damage detection method based on the data deletion model of CMD was established, as presented in this paper.

Damage detection method based on the deletion model of CMD
3.1 Regression model of CMD Ratcliffe et al. (1997) and Zhang et al. (2018) used a variety of fitting models to study the curvature modes of beams and rods and showed that cubic polynomials were the most effective to analyze the curvature modes. Accordingly, this paper adopts cubic polynomial regression to reflect the   properties of the CMD curves. Let be the node number, and be the j-th order CMD of the i-th node, . We construct a cubic polynomial regression model: where are the regression coefficients, and is the regression error, which is assumed to obey the normal distribution (Wei and Zhong, 2001). The matrix form of the regression is as follows: where The least squares estimation of is , where is called the hat matrix. Let be the fitting of , i.e.: The estimation of variance is achieved as follows: .
3.2 Quantitative evaluation of regression change q jm Let be the j-th order CMD of the m-th node. We define the model after deleting the m-th node: The matrix form is as follows: where, β(m) Thus, the least squares estimation of in Eq. (6) is as follows:β The estimation of variance is achieved as follows: .
As analyzed in Section 2.2, we can judge the influence of each node on the whole CMD curve by the vector norm . However, because its statistical distribution is uncertain, it is difficult to set a reasonable threshold to meet the need of automatic detection of civil engineering with a large amount of monitoring data. According to the statistic definition in Wei and Zhong (2001) and Chatterjee et al.(2000), we define the statistic as damage index: is the m-th diagonal member of the hat matrix . Under the assumptions described in Section 3.1, we have: The physical meaning of the damage index is to reflect the magnitude of the change in the regression curve of CMD caused by the deletion of each point. This damage index not only emphasizes the difference between regression coefficients and , but also considers the changes of residuals before and after deletion. So, it effectively reflects the m-th node influence degree on the whole curve of the CMD. Furthermore, since the distribution of is known, the damage judgment threshold can be given from the perspective of statistical analysis.

Detection of damage node
Damage detection is essentially a process of identifying a behavioral representation of a damage location that differs from the general representation. In practical engineering, for a whole structure, the damaged units represent a small proportion of the total. Therefore, the statistical characteristics of undamaged units play a dominant role in shaping the characteristics of the structure as a whole, and the damaged units are small parts that are distinguishable from most other units. Therefore, finding outliers, i.e., data inconsistent with the overall pattern, is the mathematical foundation of structural damage detection. Based on the distribution and classical probability and statistics theory, if given a significance level , the 1− confidence interval is obtained as follows: where is the upper quantile of the t-distribution with parameter . This interval is the expected fluctuation range of an undamaged node under certain random DE m disturbance. If is beyond the interval, the m-th node has a greater influence than other nodes on the regression curve of CMD and is an outlier (that is, the m-th node is the damaged node). Therefore, the evaluation criterion for the damaged node is as follows: where is called the damage detection threshold.

Discussion on choice of confidence interval α α
It can be seen from the construction process of the damage detection method based on the deletion model that the damage detection threshold in this method depends on the confidence interval, which is determined by the significance level . The significance level should be selected according to statistical theory, structural health status and a large number of engineering test results. Different structures may have different significance levels, just as in medical examination, people of different ages have different health judgment values (Mo et al., 2004).
Statistically, two types of errors can occur in automatic damage detection no matter how the interval threshold is set: (1) Type I error: the node is undamaged but detected as damaged; (2) Type II error: the node is damaged but detected as undamaged. α α α The significance level is the probability of the occurrence of type I error. Comparatively speaking, when increases, the damage detection threshold decreases, and more nodes will be judged as damaged. Instead, when decreases, the damage detection threshold increases. The damage detection is relatively severe, and only the nodes with large damage indexes can be detected as damaged. So, in the absence of a large number of engineering cases to determine the selection of significance level for large structures, the following principles can be used as references.

α
(1) According to Cowles and Davis (1982), the significance level should be between 0.05 and 0.1. If it is higher than 0.1, the probability of type I error is relatively high. If it is below 0.05, the probability of type II error is greatly increased.
(2) When the pile foundation is in short service time or in good health, the significance level can be set as a large value, and the damage detection threshold can be appropriately reduced, so as to detect the location with relatively low damage degree.
(3) When the difference between the damaged and the intact is obvious or the purpose of detection is to find the severely damaged area, the significance level can be set as a small value to appropriately raise the damage detection threshold, so as to detect the location with severe damage. α (4) For a large structure, damage detection should aim at identifying the suspected damage area and then conducting on-site detection and confirmation (such as ultrasonic and radar, etc.). A conservative significance level can be selected to ensure that the damage location is within the suspected damage area, i.e., the significance level can be set as a large value, and then the damaged area can be further detected by combining other detection methods.
According to the statistical theory and the structure considered here, a significance level of 0.05 is selected for the numerical simulation and model test.
3.5 Damage detection process based on the data deletion model According to Sections 3.1−3.4, the damage detection process based on the data deletion model is shown in Fig. 6:   Fig. 6. Damage detection process.

FEM of high-pile wharf
The numerical simulation and physical test adopt the same model, and the FEM and the numbers of the front pile are shown in Fig. 7. The high-pile wharf model is 2.05 m in length, 0.9 m in width and 1.65 m in hight. The model has three spans, and the span spacing is 0.65 m. The front and middle of the wharf consist of single straight piles, and the rear consists of a pair of inclined piles. The pile body is made of steel pipe piles, each with a diameter of 0.06 m and a thickness of 2 mm. Concrete is poured on the cross beams, longitudinal beams and slabs of the wharf. In the high-pile wharf FEM, all components adopt solid elements. The material parameters are as follows: for the concrete, the density is 2500 kg/m 3 , the elastic modulus is 3.6×10 10 Pa, and Poisson's ratio is 0.2; for the steel pipe piles, the density is 7850 kg/m 3 , the elastic modulus is 2.1×10 11 Pa, and Poisson's ratio is 0.3. This paper focuses on the damage detection of the pile body above the mud surface. As the actual displacement under the consolidation depth of the pile body is zero in actual engineering, the bottom of the pile is simplified to consolidate with 0.1 m of concrete.
In practical engineering structures, their lower-order frequencies are easier to measure than their higher order frequencies. Therefore, in this paper, the damage detection of the pile is studied through the dynamic signatures at the first-order frequency. The No. 2 pile in the front row is used to simulate damage. The length between the cap and the concrete surface of the No. 2 pile is 1.3 m, which is divided into 13 units, each 0.1 m in length. The unit and node numbers of the pile body are shown in Fig. 8. The damage is set in unit 5, and the degree of damage is set as 5%, 10%, 20%, 30%. The damage to the unit is realized by reducing its stiffness.

Results of damage detection
The CMD curves under different damage conditions are shown in Fig. 9. The new method was used to detect the damage under different damage conditions; the results are shown in Fig. 10. The damaged unit (unit 5) of the pile body was accurately detected under different damage conditions. It should be noted that, in a statistical meaning, the index DE represents the change of the overall CMD curve shape after deleting a node under a given condition. The DE values under different damage conditions are not numerically comparable. So, as shown in Fig. 10, the DE values of unit 5 under different damage conditions do not have a linear re-lationship with the damage degrees. The damage detection method based on the data deletion model of CMD realizes accurate and automatic damage detection of pile foundation in high-pile wharf.

Robustness to noise
To verify the robustness of the new method, the MAT-LAB function was used to incorporate 100 dB, 90 dB, 80 dB, 70 dB, 60 dB Gaussian white noises into the curvature mode of the FEM under each damage condition, where and denote noisy and clean curvature modes, respectively. SNR is the signal-tonoise ratio, and the option 'measured' means that the power of was measured before adding the noise. The method was used to detect damage in 1 000 experiments, which were simulated by a Monte Carlo algorithm under different SNRs of each damage condition. The success rate of damage detection was obtained, as presented in Table 3. At 100 dB, 90 dB and 80 dB SNR, the success rate was equal to or close to 100%, and at 70 dB SNR, it was larger than 90%. That is, the new method has strong robustness.

Introduction to the dynamic test model
The physical model of high-pile wharf has the same geometric dimensions, unit numbers, damage conditions and node numbers of pile body as the FEM. The upper beam, carling and panel are poured with C60 concrete, and the pile body is made of Q235 steel tube pile. The bottom of pile is fixed with 0.1 m concrete, and the bottom concrete is consolidated on the ground. The physical model is shown in Fig. 11. The damage of a unit is realized by reducing the section size.
The acceleration and strain responses under hammering load were collected for mode analysis. The frequency and mode of pile foundation were analyzed based on the acceleration responses. The similarity between the physical model and the FEM was verified by comparing their frequencies and modes (see Section 5.2). Because the curvature mode and the strain mode have the same shape (Sazonov Fig. 7. Numbers of the front piles. and Klinkhachorn, 2005), the strain mode obtained by strain response replaced curvature mode to detect damage. The acceleration vibration pick-ups adopted YD-186 piezoelectric acceleration sensor with frequency response range of 0.2− 6 kHz and sensitivity of 10±0.03 mV/ms −2 . Thirteen acceleration vibration pick-ups were arranged at 1−13 nodes, as shown in Fig. 12. Thirteen strain gauges were placed at the middle point of each unit (due to the influence of boundary conditions, the observation error of unit 1 was relatively large, so unit 1 was ignored in the subsequent analysis). In the dynamic experiment, the responses of acceleration vibration pick-ups and strain gauges were collected by DH5920 dynamic signal acquisition and analysis system, which can realize multi-channel parallel synchronous acquisition. The single channel sampling frequency was 1 kHz. The dynamic responses were analyzed by stochastic subspace identification (SSI) (Peeters and De Roeck, 1999) to obtain the frequencies, vibration modes and curvature modes of the pile foundation.
5.2 Similarity between the physical model and the FEM The frequency comparison between the experiment and the FEM is shown in Table 4. The experimental frequencies are very close to the FEM simulation, and the maximum relative error is smaller than 4%. The similarity between the experiment mode and the FEM mode is shown in Fig.13 and Fig. 14 (due to space limitations, we present only the mode    comparison of intact and 10% damage of the No. 2 pile). The experiment modes and the FEM modes are very similar; i.e., the dynamic signatures of the experiment and the FEM are consistent.

Application to the experimental model
The CMDs under different conditions were obtained and are shown in Fig. 15. Owing to the existence of experimental noise, boundary conditions, mode analysis error and other reasons, there are differences between the experiment CMD and the FEM CMD. Such differences are a common problem in current dynamic experiments (Cao et al., 2016). Fig. 16 shows the damage detection results of each condition. The damage unit (unit 5) has been accurately identified under each condition. The experimental verification confirms that the new method has good practicability even   under large experimental errors.

Conclusions
(1) The curve characteristic of the CMD to structure damage is summarized. The cubic polynomial regression model is established and the influence of data points on the overall curve is measured by the data deletion model. The results show that the position of inverted 'V' shape of the curve can be accurately detected by the data deletion model.
(2) The DE-statistic to evaluate the change of regression coefficients after deleting a point of the CMD is constructed, so as to quantitatively analyze the relative influence at each node on the overall CMD curve shape. Combined with the significance level and confidence interval, the automatic damage detection of high-pile wharf is realized. And the choice of significance level is discussed according to the actual condition of the engineering.
(3) The FEM of dynamic damage to high-pile wharf is established, and location of the damage in wharf foundation pile is accurately detected based on the new method, which proves the effectiveness of this method. The reliability of the method is verified by calculating the success rate of damage detection of the FEM under different intensity of Gaussian white noise.
(4) The physical experimental model of a high-pile wharf is established and the curvature modes under different damage conditions are measured. The new method is applied to damage detection for the physical experimental models, and the results show that the new method has strong practicability. The study provides a reference for the damage detection of pile foundation in high-pile wharf and similar projects.