# Structural Damage Localization and Quantification Based on a CEEMDAN Hilbert Transform Neural Network Approach: A Model Steel Truss Bridge Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Motivation

## 2. Theoretical Background

#### 2.1. Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

_{j}[w

_{i}(t)] at each step of the decomposition process, instead of adding the white Gaussian noise that is obtained with a unique residue after the extraction of each IMF. The CEEMDAN method can be defined as the following steps:

- Add E1(wi(t)) (i = 1, 2, …, I), to the initial signal, x(t), where wi, β, and I indicate the ith added white Gaussian noise, the amplitude of the ith added white noise, and ensemble size, respectively:x
_{i}(t) = x(t) + β_{0}E_{1}(w_{i}(t)) - Calculate the first IMF (${\overline{c}}_{1}\left(t\right)$) through the first residue (i.e., r
_{1}(t)) as follows:$${\overline{c}}_{1}(t)=x(t)-{r}_{1}(t),\text{}\mathrm{where}\text{}{r}_{1}(t)=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}M({x}_{i}}(t))$$ - Obtain the second IMF ${\overline{c}}_{2}(t)={r}_{1}(t)-{r}_{2}(t)$, where ${r}_{2}(t)=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}M\left({r}_{1}(t)+{\beta}_{1}{E}_{2}({w}_{i}(t))\right)}$, and E
_{2}(w_{i}(t)) is the second IMF of EEMD. - Repeat Step 3 to obtain jth IMF of CEEMDAN ${\overline{c}}_{j}(t)={r}_{j-1}(t)-{r}_{j}(t)$ where$${r}_{j}(t)=\frac{1}{I}{\displaystyle \sum _{i=1}^{I}M\left({r}_{j-1}(t)+{\beta}_{j-1}{E}_{j}({w}_{i}(t))\right)}$$
_{j}= ε_{0}std[r_{j}(t)] is the signal-noise ratio (SNR).

#### 2.2. Hilbert–Huang Transform (HHT)

#### 2.3. Applications of ANNs

#### 2.4. Damage Indices Based on the CEEMDAN-HT-ANN Model

_{Healthy}and E

_{Damaged}represent the output of the CEEMDAN-HT-ANN model based on the acceleration response of the healthy and damaged states, respectively.

_{IMF}are used as the input and target layers of a healthy state of the truss in the training process of the ANN, respectively. Then, the IMFs of the damage scenarios are used to test the ANN. Accordingly, the damage index based on IA can be defined as follows:

_{Healthy}and IA

_{Damaged}represent the output of the CEEMDAN-HT-ANN model based on the acceleration response of the healthy and damaged states, respectively.

_{IMF}are used as input and target layers of the healthy state of the truss in the training process of the ANN, respectively, then the IMFs of the damage scenarios are used to test the ANN. Accordingly, the damage index based on P can be defined as follows:

_{Healthy}and P

_{Damaged}represent the output of the CEEMDAN-HT-ANN model based on the acceleration response of the healthy and damaged states, respectively.

## 3. Applications

#### 3.1. Experimental Setup

#### 3.2. Procedure of the CEEMDAN-HHT-ANN Damage Detection Method

## 4. Result and Discussion

#### 4.1. Detection of the Presence and Severity of Damage

_{1}, T

_{2}, and T

_{3}are 0.051, 0.025, and 0.016 s, respectively. Hence, considering 1.0 s of the acceleration response of the bridge (about 20T

_{1}) is a reasonable time window to analyze in this study.

#### 4.2. Detection of Damage Location

## 5. Summary and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A perspective view of the steel truss bridge model as the case study in the present work established in the Qingdao University of Technology (QUT) laboratory.

**Figure 3.**A schematic view of the truss bridge model by showing the uniform distribution of the locations of sensors, and the location of the damaged element.

**Figure 4.**Details of the damaged elements with; (

**a**) 20%, (

**b**) 50%, and (

**c**) 80% reduced percentages of the cross-section stiffness.

**Figure 5.**The framework flowchart of the proposed CEEMDAN-HT-ANN model to detect, locate, and classify the severity of the damage.

**Figure 6.**(

**a**) acceleration response of sensor-10 for a healthy state of the bridge; (

**b**) the first three intrinsic mode functions (IMFs) extracted by the CEEMDAN for sensor-10 for the healthy state and different damage levels.

**Figure 7.**Evaluation of the accuracy performance of training the Artificial Neural Network (ANN) based on the energy parameter of sensor-10 for the healthy state of the truss. (where R, µ, σ, MSE, and RMSE denotes the value of regression, mean, standard division, mean square error, and root mean square error, respectively).

**Figure 8.**Comparison of the IMFs’ energy of sensor-10 between the outputs of the trained model (i.e., estimated) and the measured data for 100% damage state of the truss.

**Figure 9.**The energy of IMFs of the sensor-10 estimated by the proposed model for the healthy state and different damage levels of the truss.

**Figure 10.**The estimated instantaneous amplitude (IA) of the sensor-10 by the proposed model for the healthy state and different damage levels of the truss.

**Figure 11.**Comparison of the IMFs’ unwrapped phase of acceleration responses of sensor-10 estimated by the proposed model for the healthy and different damage levels the truss.

**Figure 12.**Spectrograms output of the IMFs of sensor-10 under five states of the structure including; (

**a**) healthy; (

**b**) 20% damage; (

**c**) 50% damage; (

**d**) 80% damage; and (

**e**) 100% damage by the CEEMDAN-HT-ANN model.

**Figure 13.**The energy output of the IMFs for different sensor locations from the damaged element by the proposed model.

**Figure 14.**The IA output of the IMFs for different locations from the damaged element by the proposed model.

**Figure 15.**Comparison of the IMFs’ unwrapped phase outputs of the acceleration responses by the proposed model for different sensor locations from the damaged element.

**Figure 16.**Spectrograms output of the IMFs for different locations of (

**a**) the nearest, (

**b**) second-nearest, (

**c**) far, and (

**d**) the farthest sensors from the 100% damaged element through the proposed model.

**Table 1.**Damage index (DI) results based on the energy, instantaneous amplitude (IA), and unwrapped phase of IMFs for the different damage levels.

IMF Feature | Damage 20% | Damage 50% | Damage 80% | Damage 100% |
---|---|---|---|---|

DI (%) | DI (%) | DI (%) | DI (%) | |

Energy | 23.43 | 32.72 | 86.47 | 94.24 |

IA | 21.46 | 25.88 | 57.39 | 62.04 |

Unwrapped phase | 6.52 | 17.13 | 20.11 | 41.80 |

**Table 2.**Damage index results based on the IMF features for different sensor locations from the damaged element.

IMF Feature | Nearest | Second Nearest | Far | Farthest |
---|---|---|---|---|

DI (%) | DI (%) | DI (%) | DI (%) | |

Energy | 94.24 | 35.84 | 24.92 | 17.34 |

IA | 62.04 | 51.36 | 31.35 | 16.45 |

Unwrapped Phase | 41.80 | 18.90 | 07.55 | 05.10 |

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## Share and Cite

**MDPI and ACS Style**

Mousavi, A.A.; Zhang, C.; Masri, S.F.; Gholipour, G. Structural Damage Localization and Quantification Based on a CEEMDAN Hilbert Transform Neural Network Approach: A Model Steel Truss Bridge Case Study. *Sensors* **2020**, *20*, 1271.
https://doi.org/10.3390/s20051271

**AMA Style**

Mousavi AA, Zhang C, Masri SF, Gholipour G. Structural Damage Localization and Quantification Based on a CEEMDAN Hilbert Transform Neural Network Approach: A Model Steel Truss Bridge Case Study. *Sensors*. 2020; 20(5):1271.
https://doi.org/10.3390/s20051271

**Chicago/Turabian Style**

Mousavi, Asma Alsadat, Chunwei Zhang, Sami F. Masri, and Gholamreza Gholipour. 2020. "Structural Damage Localization and Quantification Based on a CEEMDAN Hilbert Transform Neural Network Approach: A Model Steel Truss Bridge Case Study" *Sensors* 20, no. 5: 1271.
https://doi.org/10.3390/s20051271