# Seismic Damage Identification Method for Curved Beam Bridges Based on Wavelet Packet Norm Entropy

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## Abstract

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## 1. Introduction

_{p}norm characteristic of wavelet coefficients, and the information characteristic is measured quantitatively by the information entropy. The damage index therefrom combines the advantages of WPT, L

_{p}norm, and information entropy, which greatly improves the sensitivity and accuracy of damage identification.

_{p}norm to extract damage information, endowing WPNE-based methods with new damage characterization capabilities. In addition, WPNE adds a valid p value selection process compared with WPSE, which strengthens the damage distinction ability of the index. In terms of anti-noise performance, the WPNE-based method has stronger noise immunity.

_{p}norm and information entropy. Section 3 constructs a damage index based on WPNE and introduces the identification steps of CCRFBs. Section 4 establishes a finite element model of the CCRFB and carries out the dynamic analysis of the CCRFB. The identification results of the CCRFB are in presented in Section 5. Section 6 analyzes the effect of seismic excitation and compares the identification performance based on the WPNE and tests the noise resistance of the damage index. The conclusions are presented in Section 7.

## 2. Basic Theory

#### 2.1. Wavelet Packet Transform

^{j}bands, some specific bands (especially the high-frequency bands) conceal information on individual instances of damage.

#### 2.2. L_{p} Norm

_{p}norm. If ${X=[x}_{1}{,x}_{2}{,x}_{3}{,\dots ,x}_{n}{]}^{T}$, the L

_{p}norm of vector $X$ can then be written as [44]:

_{p}norm. Values of 1, 2 or $+\infty $ are often taken as the p value. When p = 1, the norm L

_{1}is also known as the Manhattan Distance, which is capable of characterizing the difference between different vectors and clearing the features without information and meaning, thereby achieving the sparsity of the vector. When p = 2, the norm L

_{2}is also known as the Euclidean Distance, which can also express the difference between different vectors, but L

_{2}is generally used to optimize the regularization item of the target function. The ${L}_{\infty}$ norm is able to extract the largest element in the vector.

#### 2.3. Information Entropy

## 3. Damage Identification Method

#### 3.1. Damage Identification Index

_{p}norm of ${c}_{j}^{i}\left(t\right)$, its expression is:

_{p}norm and information entropy, which are embodied in: (1) the high-resolution characteristics of the WPT are used to achieve more detailed decomposition of nonlinear and non-stable response signals, highlighting the information of the full band; (2) L

_{p}norm is applied to the abstract effective damage characteristics and the sparse useless interference feature; (3) the degree of uncertainty of the information system is quantitatively characterized by information entropy. In summary, WPNE is capable of highlighting detail, extracting features and quantifying information. Based on WPNE, we can construct the structural damage identification index ${DI}_{WPNE}$; the index is defined as follows:

^{h}and WPNE

^{d}are WPNE in the state of structural health and damage, respectively. According to Formula (11), DI

_{WPNE}reflects the relative amount of the change before and after the damage, moreover the size of the change represents different states of the structure, that is, health or damage state. When the structure is not damaged or is slightly damaged, DI

_{WPNE}is zero or close to zero and the DI

_{WPNE}curve is relatively flat. When the structure is damaged to a certain extent, the value of DI

_{WPNE}at the damage position is a positive number greater than zero, and the DI

_{WPNE}curve shows a significant mutation at the damage position, displaying a sudden peak. The greater the severity of the damage, the greater the value of the peak. Therefore, the DI

_{WPNE}index is capable of identifying the location of the damage and characterizing the severity of the damage.

_{WPNE}is established on the basis of the difference between the DI

_{WPNE}index and the damage threshold ${\mathit{DI}}_{\mathit{WPNE}}^{\mathit{TH}}$:

_{WPNE}index value of all measurement points, respectively; ${u}_{\alpha}$ is the upper $\alpha $ quantile of the standard normal distribution, $\alpha $ is commonly referred to as the significance level, which generally takes on 0.05, 0.02, 0.015 or other small values, (1−$\alpha $) is called confidence probability, where $\alpha $ has a value of 0.02 in this paper, and the corresponding confidence probability is 98%. By checking the upper $\alpha $ quantile table of the standard normal distribution, ${u}_{0.02\text{}}=2.06$ can be found.

_{WPNE}index value of the measurement point is greater than the damage threshold ${\mathit{DI}}_{\mathit{WPNE}}^{\mathit{TH}}$, EW

_{WPNE}> 0, the health monitoring system will 98% believe that damage will appear at the measurement point, and the system will provide an early warning; (2) when the DI

_{WPNE}index value of the measurement point is not greater than the damage threshold ${\mathit{DI}}_{\mathit{WPNE}}^{\mathit{TH}}$, ${\mathit{EW}}_{\mathit{WPNE}}\text{}\le \text{}0$, the health monitoring system will 98% believe that the structure is not damaged, and the system will not provide a warning. Theoretically, a zero value of EW

_{WPNE}can be used as the damage warning value. However, in the course of practical application, in order to reduce false reports caused by environmental factors (e.g., temperature, noise, and humidity), initial micro-damage, and measurement errors, numbers greater than zero are usually taken as the damage warning value.

#### 3.2. Damage Identification Steps

_{WPNE}index, introduce the damage threshold ${\mathit{DI}}_{\mathit{WPNE}}^{\mathit{TH}}$ to further obtain the damage warning index EW

_{WPNE}, determine the damage position according to the sudden peak of the index curve, and identify the severity of damage using the peak value.

_{er}(wavelet packet energy ratio change rate index). Add white Gaussian Noise with different signal-to-noise ratios to analyze the noise resistance of the damage identification index.

## 4. Dynamic Analysis of CCRFB

#### 4.1. Establish a CCRFB Finite Element Model

#### 4.2. Set Damage Scenarios

#### 4.3. Enter Ground Motion Acceleration

#### 4.4. Measured Dynamic Response

## 5. Damage Identification for CCRFB

#### 5.1. Choose Optimal Dynamic Response

_{x}and U

_{y}response data for each point of the CCRFB were extracted. The No. 63 measurement point of scenario 1 is taken as an example; Figure 7 presents the U

_{x}and U

_{y}response of the measurement point and the corresponding frequency spectrum. Obviously there is no significant difference between U

_{x}and U

_{y}response in the time frequency domain, so it is hard to judge the sensitivity of U

_{x}and U

_{y}response to damage. To select a displacement response that is more sensitive to damage, according to the existing damage identification index D

_{er}(wavelet package energy ratio change rate), the damage identification effects of U

_{x}and U

_{y}response under scenario 2 are compared. For a definition of D

_{er}, please see Formula (14) [47]. For the results of D

_{er}index damage identification based on U

_{x}and U

_{y}response, please see Figure 8.

_{er}index value of the damage identification based on U

_{x}and U

_{y}response is larger at the damage location, and the D

_{er}curve shows a significant mutation in the damage area with an obvious peak, but it is relatively flat and smooth in other non-damaged positions. (2) The D

_{er}index value of damage identification based on U

_{y}response at the No. 66 measurement point of the damage is not much different from that of the D

_{er}index value at the non-damaged measurement point. (3) Compared with the ${U}_{y}$ response, damage identification based on the U

_{x}response is more prominent at the damage position. On the whole, the damage identification effect based on the U

_{x}response is better. Therefore, the U

_{x}response is selected below for structural damage identification research.

#### 5.2. Select Optimal Wavelet Packet Parameters

_{x}response extracted from the No. 63 measurement point under scenario 1 under SF seismic excitation is used to calculated the M value of the base function to be determined, with a decomposition scale of 2–10. The results are shown in Figure 9, on the basis of which we can see that the M value of sym13 in all the wavelet packet basis functions to be determined is relatively small overall, so it is necessary to choose the sym13 wavelet as the optimal wavelet basis function, compare the identification accuracy of the sym13 wavelet at different scales, and consider the calculation efficiency; finally, an optimal decomposition scale of 7 was chosen.

#### 5.3. Select Valid p Values

_{p}norm is introduced in the construction of WPNE. According to Formula (8), the damage identification indexes based on WPNE can be further determined only by selecting the appropriate p value, but there is no uniform standard for the determination of p values. When p is 2, the L

_{2}norm of the band is essentially the square root of the wavelet package energy. Theoretically, indexes of p greater than or equal to 1 are valid, but the larger the p value is, the longer the corresponding calculation time will be. Therefore, under the premise of ensuring the accuracy of damage identification, smaller values should be taken for p to improve calculation efficiency. To select the most effective p values, it is necessary to compare the damage identification effect under the conditions of scenario 2 when $p=0.2\text{}\times \text{}n+0.8\text{}(n=1,2,3,\dots ,11)$. A comparison of the damage identification results is provided in Figure 10, on the basis of which it can easily be found that when $p\text{}\ge \text{}1$, DI

_{WPNE}shows a mutation in the damage position, with a peak; DI

_{WPNE}is always able to accurately identify the location of the damage. Nevertheless, the larger the p value, the smaller the peak value will be, and the longer the calculation time will be. Hence, it is necessary to comprehensively consider calculation efficiency, damage identification sensitivity, and damage positioning accuracy, and thus 1 is selected as a valid p value. The following analysis is based on WPNE when p = 1.

#### 5.4. Damage Identification Results

_{x}response measured at the No. 63–93 measurement points on bridge pier No. 3 of the CCRFB before and after the occurrence of damage is decomposed by the wavelet packet, then DI

_{WPNE}is calculated when p is equal to 1, and the damage early warning index EW

_{WPNE}is further calculated in combination with the damage threshold ${\mathit{DI}}_{\mathit{WPNE}}^{\mathit{TH}}$. Figure 11 shows the identification results of the CCRFB under SF seismic excitation. In the picture, scenarios 2–8 incorporate lower damage I, and scenarios 9–15 incorporate upper damage II.

_{WPNE}is greater than zero at the damage position, with a peak, and is lower than zero at non-damaged locations. Furthermore, EW

_{WPNE}increases as the severity of damage increases; therefore, EW

_{WPNE}can accurately locate the lower damage I and characterize the severity of the damage. For upper damage II, although EW

_{WPNE}is greater than zero at the location of the damage, with a peak, EW

_{WPNE}is also greater than zero in some non-damaged positions close to the damage. Accordingly, EW

_{WPNE}is not suitable for identifying upper damage II.

_{WPNE}index to identify upper damage II, it is necessary to take into account the peak at the upper damage measurement point and good curve continuity in non-damaged areas. According to the central difference principle, the upper damage correction index SEW

_{WPNE}is constructed to magnify the curve mutation and remove the holistic trend of the curve, further highlighting the damage position. The definition of SEW

_{WPNE}is shown in Formula (16), below, and the damage identification results of upper damage II using the correction index SEW

_{WPNE}are shown in Figure 12.

_{WPNE}essentially refers to the second-order numerical differentiation of EW

_{WPNE}, through which the singular point of EW

_{WPNE}is prominent in the form of numerical differentiation. Formula (16) can only calculate the SEW

_{WPNE}value of non-endpoints. For the No. 63 and No. 93 measurement points at the endpoint, Formulas (17) and (18) are used to calculate the SEW

_{WPNE}value.

_{WPNE}is only greater than 0 at damage measurement points (30, 31), and increases with increasing damage severity, while ${\mathit{SEW}}_{\mathit{WPNE}}\text{}\le \text{}0$ at other non-damaged measurement points. Therefore, for upper damage II, the SEW

_{WPNE}index has a strong capacity for damage identification and positioning, as well as the ability to characterize the severity of structural damage. Unfortunately, due to the introduction of the second-order central difference algorithm to the calculation of SEW

_{WPNE}, computational noise may be introduced, thereby reducing the immunity of the SEW

_{WPNE}index to noise interference. The noise resistance of the EW

_{WPNE}and SEW

_{WPNE}indexes will be discussed in detail in the next section.

_{WPNE}and SEW

_{WPNE}index values at the 5% damage severity are taken as the damage warning value for damage I and II, respectively. The damage warning values for the CCRFB are shown in Table 2.

_{WPNE}index value of the lower damage and SEW

_{WPNE}index value of the upper damage are not on the same order of magnitude. Furthermore, Table 2 shows that the value of the lower damage warning is about 60 times that of the upper damage warning value, so it is necessary to separate the upper damage identification from the lower damage identification and to consider them individually. The damage warning value of Table 2 is applied to the identification of all the CCRFB bridge piers (P1–P6). The damage I identification results are shown in Figure 13, and the damage II identification results are shown in Figure 14. The black dashed line in the figure refers to the damage warning value. According to Section 4.2, the damage position of the CCRFB is only set on the P3 pier, the serial numbers of the measurement points for damage I are (64, 65, 66), and the serial numbers of the measurement points for damage II are (92, 93). It is not difficult to see from the figure that both EW

_{WPNE}and SEW

_{WPNE}at the damage measurement point of the P3 pier exceed the damage warning value, and the damage index value increases with increasing damage severity. In other non-damaged piers, EW

_{WPNE}and SEW

_{WPNE}do not exceed the damage warning value, which is consistent with the actual situation. In summary, the WPNE-based damage identification index can accurately identify the damage position of the CCRFB and quantitatively characterize the severity of damage, thereby achieving structural damage detection, positioning and early warning.

## 6. Discussion

#### 6.1. Compare Identification Index D_{er}

_{er}index. The definition of D

_{er}is detailed in Formula (14) in Section 5.1. D

_{er}identifies the damage position by means of the energy change of each band before and after the damage takes place. Similarly, according to Formulas (12) and (13), the damage threshold based on D

_{er}is set to further obtain the damage warning index ${\mathit{EW}}_{\mathit{er}}$. The identification results of ${\mathit{EW}}_{\mathit{er}}$ are shown in Figure 15. The identification results of ${\mathit{EW}}_{\mathit{er}}$ resemble those of EW

_{WPNE}. ${\mathit{EW}}_{\mathit{er}}$ can accurately identify lower damage I in the CCRFB. For upper damage II, the area where ${\mathit{EW}}_{\mathit{er}}\text{}$ 0 is far beyond the damage measurement points, and ${\mathit{EW}}_{\mathit{er}}$ possesses a poor ability of detect damage II. Likewise, in accordance with the characteristics of the ${\mathit{EW}}_{\mathit{er}}$ curve changes, Formulas (16)–(18) were used to construct a correction index ${\mathit{SEW}}_{\mathit{er}}$ and introduce the damage warning value. ${\mathit{EW}}_{\mathit{er}}$ and ${\mathit{SEW}}_{\mathit{er}}$ were applied in the damage identification for all bridge piers in the CCRFB. The results of damage I identification with ${\mathit{EW}}_{\mathit{er}}$ are shown in Figure 16, and the results of damage II identification with ${\mathit{SEW}}_{\mathit{er}}$ are shown in Figure 17.

_{er}both show large mutations at the damage position, and the index values at the damage position are all greater than the damage warning values. However, the index values at non-damaged bridge pier boundary (the solidification of the pier bottom) and the pier-beam connection are also greater than the damage warning values, which does not conform to the actual situation, so the damage identification effect of ${\mathit{EW}}_{\mathit{er}}$ and ${\mathit{SEW}}_{\mathit{er}}$ is not good. In summary, compared with D

_{er}, the index based on WPNE has a stronger ability of damage identification and higher damage positioning accuracy.

#### 6.2. Noise Resistance Analysis

_{x}response measured in a noise-free environment is superimposed with zero white Gaussian noise. The U

_{x}response containing the noise is used for CCRFB damage identification. The noise level is measured by the physical signal-to-noise ratio ($\mathit{SNR}$), defined as follows [50]:

_{x}response of the CCRFB measured at the No. 63 measurement point under scenario 2 is taken as an example. Figure 18 shows the time domain and frequency spectrum before and after superimposing 60 dB noise onto the U

_{x}response of the measurement point. The figure shows that there is no obvious difference in the U

_{x}response before and after adding noise in the time domain, but there are significant differences in the frequency domain, especially in the high-frequency band (12–24 Hz). Because white Gaussian noise is a random noise, one test cannot fully evaluate the noise resistance of the damage index. Therefore, 10 separate tests were carried out under the same noise intensity, and the average of 10 test results was taken as the damage identification result under each noise level. Owing to the fact that there is a numerical difference between EW

_{WPNE}and SEW

_{WPNE}of an order of magnitude, the noise resistance of EW

_{WPNE}and SEW

_{WPNE}will be discussed separately below.

_{WPNE}and SEW

_{WPNE}damage identification in an environment with 60 dB of noise. It can be seen from the figure that EW

_{WPNE}is able to accurately identify the lower damage location in a noisy environment where $SNR=60\mathrm{dB}$, with a strong noise robustness; however, for the upper damage identification, SEW

_{WPNE}suffers from many false reports and missing reports. This means that SEW

_{WPNE}with the interference of 60 dB of noise is unable to precisely locate the damage, offering poor noise resistance. The reason for this is that the construction of SEW

_{WPNE}is combined with the central difference operator (Formula (16)), indirectly introducing computational noise, thereby reducing the noise immunity SEW

_{WPNE}.

_{x}response (the No. 63 measurement point under scenario 2) before and after adding 60 dB noise. According to Figure 18c and Figure 20, the noise signal, which generally has low energy and high frequency, mainly affects the high-frequency part of the response signal. After WPT, the effective information of the response is mainly in the low-frequency band, and noise information is mainly distributed in the high-frequency band, that is, the noise has a larger impact on wavelet coefficients of the high-frequency band. Therefore, selecting the top n low-frequency bands with high energy for the calculation of WPNE can avoid noise interference to a certain extent. According to Formulas (9) and (10), WPNE of the top n low-frequency bands can be expressed as:

#### 6.3. Effect of Seismic Excitation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Geometric dimensions of the CCRFB: (

**a**) plane graph; (

**b**) elevation graph; (

**c**) main girder cross section.

**Figure 4.**Input seismic excitation: (

**a**) the most unfavorable input direction; (

**b**) SF accelerogram; (

**c**) Fourier spectrum of SF.

**Figure 6.**Displacement response of No. 64 measurement point: (

**a**) scenario 1; (

**b**) scenario 2; (

**c**) the response difference.

**Figure 7.**Displacement response of No. 63 measurement point under scenario 1: (

**a**) U

_{x}; (

**b**) U

_{y}; (

**c**) Fourier spectrum.

**Figure 18.**Time–frequency diagram before and after adding 60 dB noise: (

**a**) no noise signal; (

**b**) additional 60 dB noise signal; (

**c**) frequency spectrum.

**Figure 19.**The identification results of the damage indexes when SNR = 60 dB: (

**a**) EW

_{WPNE}; (

**b**) SEW

_{WPNE}.

**Figure 23.**Damage identification results of the CCRFB under WN seismic excitation: (

**a**) EW

_{WPNE}; (

**b**) SEW

_{WPNE}.

Damage Location | Stiffness Reduction Rate | Damage Scenarios | Damage Location | Stiffness Reduction Rate | Damage Scenarios |
---|---|---|---|---|---|

I, II | 0% | 1 | |||

I | 5% | 2 | II | 5% | 9 |

I | 10% | 3 | II | 10% | 10 |

I | 15% | 4 | II | 15% | 11 |

I | 20% | 5 | II | 20% | 12 |

I | 25% | 6 | II | 25% | 13 |

I | 30% | 7 | II | 30% | 14 |

I | 35% | 8 | II | 35% | 15 |

Lower Damage I (EW _{WPNE}) | Upper Damage II (SEW_{WPNE}) | |
---|---|---|

Damage warning value | 3.8000 × 10^{−2} | 5.9645 × 10^{−4} |

Scenario | MRR(%) | FRR(%) | NII(%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

SNR(dB) | SNR(dB) | SNR(dB) | ||||||||||

20 | 30 | 40 | 50 | 20 | 30 | 40 | 50 | 20 | 30 | 40 | 50 | |

2 | 8.92 | 2.03 | 0.00 | 0.00 | 14.42 | 7.76 | 2.56 | 0.00 | 77.95 | 90.37 | 97.44 | 100.00 |

3 | 5.21 | 1.70 | 0.00 | 0.00 | 10.27 | 5.47 | 0.80 | 0.00 | 85.06 | 92.92 | 99.20 | 100.00 |

4 | 1.09 | 0.04 | 0.00 | 0.00 | 8.71 | 3.28 | 0.10 | 0.00 | 90.29 | 96.68 | 99.90 | 100.00 |

5 | 0.02 | 0.00 | 0.00 | 0.00 | 3.55 | 1.29 | 0.00 | 0.00 | 96.43 | 98.71 | 100.00 | 100.00 |

6 | 0.00 | 0.00 | 0.00 | 0.00 | 1.08 | 1.06 | 0.00 | 0.00 | 98.92 | 98.94 | 100.00 | 100.00 |

7 | 0.00 | 0.00 | 0.00 | 0.00 | 0.53 | 0.25 | 0.00 | 0.00 | 99.47 | 99.75 | 100.00 | 100.00 |

8 | 0.00 | 0.00 | 0.00 | 0.00 | 0.09 | 0.00 | 0.00 | 0.00 | 99.91 | 100.00 | 100.00 | 100.00 |

Scenario | MRR(%) | FRR(%) | NII(%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

SNR(dB) | SNR(dB) | SNR(dB) | ||||||||||

40 | 50 | 60 | 70 | 40 | 50 | 60 | 70 | 40 | 50 | 60 | 70 | |

9 | 12.34 | 4.55 | 1.07 | 0.48 | 17.29 | 3.34 | 2.45 | 1.12 | 72.50 | 92.26 | 96.51 | 98.40 |

10 | 9.28 | 2.29 | 0.02 | 0.00 | 13.70 | 1.46 | 0.43 | 0.20 | 78.29 | 96.28 | 99.55 | 99.80 |

11 | 7.90 | 1.15 | 0.00 | 0.00 | 10.55 | 0.83 | 0.02 | 0.00 | 82.38 | 98.03 | 99.98 | 100.00 |

12 | 3.89 | 0.05 | 0.00 | 0.00 | 7.01 | 0.46 | 0.00 | 0.00 | 89.37 | 99.49 | 100.00 | 100.00 |

13 | 0.77 | 0.00 | 0.00 | 0.00 | 4.96 | 0.01 | 0.00 | 0.00 | 94.31 | 99.99 | 100.00 | 100.00 |

14 | 0.06 | 0.00 | 0.00 | 0.00 | 3.01 | 0.00 | 0.00 | 0.00 | 96.93 | 100.00 | 100.00 | 100.00 |

15 | 0.00 | 0.00 | 0.00 | 0.00 | 2.11 | 0.00 | 0.00 | 0.00 | 97.89 | 100.00 | 100.00 | 100.00 |

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## Share and Cite

**MDPI and ACS Style**

Deng, T.; Huang, J.; Cao, M.; Li, D.; Bayat, M.
Seismic Damage Identification Method for Curved Beam Bridges Based on Wavelet Packet Norm Entropy. *Sensors* **2022**, *22*, 239.
https://doi.org/10.3390/s22010239

**AMA Style**

Deng T, Huang J, Cao M, Li D, Bayat M.
Seismic Damage Identification Method for Curved Beam Bridges Based on Wavelet Packet Norm Entropy. *Sensors*. 2022; 22(1):239.
https://doi.org/10.3390/s22010239

**Chicago/Turabian Style**

Deng, Tongfa, Jinwen Huang, Maosen Cao, Dayang Li, and Mahmoud Bayat.
2022. "Seismic Damage Identification Method for Curved Beam Bridges Based on Wavelet Packet Norm Entropy" *Sensors* 22, no. 1: 239.
https://doi.org/10.3390/s22010239