# Guided Wave-Based Damage Detection of Square Steel Tubes Utilizing Structure Symmetry

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Propagation Characteristics and Symmetry Analysis of Guided Waves in Square Steel Tube

#### 2.1. Dispersion Characteristics Analysis Based on SAFE Method

**K**

_{1},

**K**

_{2}and

**K**

_{3}are the different stiffness matrices,

**M**is the mass matrix, the subscript N is the number of degrees of freedom, and

**U**is the nodal displacement vector. In the above equation, there are two unknown variables: wave number ξ and circular frequency ω. With a given circular frequency, the real solution ξ

_{Re}of finite wave numbers can be obtained by solving the eigenvalues of the equation. Equation (1) can be rewritten as:

_{th}mode can be further obtained by using the following equations:

^{3}, 205 GPa, and 0.28, respectively. The cross-section of the square steel tube is discretized into 426 triangular elements and 324 nodes, as shown in Figure 2.

#### 2.2. Symmetry Analysis for Guided Waves in Square Steel Tube

**T**and

**T**

_{n}(y, z) denote the actual stress field and the stress field of the nth mode, and α

_{n}(x) is the combination coefficient for the nth mode which depends on the specific excitation situation.

#### 2.3. The Procedure of the Damage Detection Method

_{D}is ToA of the initial wave, and T

_{E}is ToA of the damage reflected wave.

## 3. Numerical Simulation

#### 3.1. Propagation Characteristic of Guided Waves in Square Steel Tube

_{c}is the excitation frequency of the signal; and n is the period number of the signal.

#### 3.2. Damage Detection

## 4. Experimental Investigations

#### 4.1. Experimental Setup

#### 4.2. Inspection of Different Shape Damage

#### 4.3. Damage Localization

#### 4.4. Further Discussion on Corner Damage and Multiple Damages

## 5. Conclusions

- The SAFE method was used to solve the dispersion characteristics of the square steel tube, and the first, second, fifth and sixth order modes were selected as the desired modes for inspecting the damage status of long and short edges. Through finite element simulation and NME analysis, it was verified that by using structure symmetry, the influence of the dispersion effect and boundary limit can be reduced effectively, and thus clutter can be eliminated.
- The method to identify the damage of square steel tube based on section symmetry was proposed and verified by numerical simulation and laboratory experiments. The test results show that for cracks of different sizes and middle and offset hole damage in long and short edges, the symmetrical damage difference signals will appear as obvious damage wave packets at the damage locations.
- Damage location can be determined by the group wave velocity of desired modes and the appearance time of the damage wave packet. The calculation results show that the identification errors of crack, middle hole and offset hole damage are no more than 9.11%, indicating high positioning accuracy.
- For corner hole damage, the initial scheme had a poor identification effect, but it can be improved by changing the position of the receiving piezoelectric plate. However, for multi damages, the proposed method still has a good identification effect.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Wan, X.; Liu, M.R.; Zhang, X.H.; Fan, H.W.; Tse, P.W.; Dong, M.; Wang, X.; Wei, H.H.; Xu, C.H.; Ma, H.W. The Use of Ultrasonic Guided Waves for the Inspection of Square Tube Structures: Dispersion Analysis and Numerical and Experimental Studies. Struct. Health Monit.
**2021**, 20, 58–73. [Google Scholar] [CrossRef] - He, J.Z.; Yuan, F.G. Damage Identification for Composite Structures Using a Cross-Correlation Reverse-Time Migration Technique. Struct. Health Monit.-Int. J.
**2015**, 14, 558–570. [Google Scholar] [CrossRef] - He, J.Z.; Leckey, C.A.C.; Leser, P.E.; Leser, W.P. Multi-Mode Reverse Time Migration Damage Imaging Using Ultrasonic Guided Waves. Ultrasonics
**2019**, 94, 319–331. [Google Scholar] [CrossRef] [PubMed] - He, J.Z.; Rocha, D.C.; Sava, P. Guided Wave Tomography Based on Least-Squares Reverse-Time Migration. Struct. Health Monit.-Int. J.
**2019**, 19, 1237–1249. [Google Scholar] [CrossRef] - Zhou, W.S.; Li, H.; Yuan, F.G. Fundamental Understanding of Wave Generation and Reception Using d36 Type Piezoelectric Transducers. Ultrasonics
**2015**, 57, 135–143. [Google Scholar] [CrossRef] - Golub, M.V.; Doroshenko, O.V.; Arsenov, M.A.; Bareiko, I.A.; Eremin, A.A. Identification of Material Properties of Elastic Plate Using Guided Waves Based on the Matrix Pencil Method and Laser Doppler Vibrometry. Symmetry
**2022**, 14, 1077. [Google Scholar] [CrossRef] - Yaylaci, M. Simulate of Edge and an Internal Crack Problem and Estimation of Stress Intensity Factor Through Finite Element Method. Adv. Nano Res.
**2022**, 12, 405–414. [Google Scholar] - Vinh, P.V.; Avcar, M.; Belarbi, M.O.; Tounsi, A.; Huy, L.Q. A New Higher-Order Mixed Four-Node Quadrilateral Finite Element for Static Bending Analysis of Functionally Graded Plates. Structures
**2023**, 47, 1595–1612. [Google Scholar] [CrossRef] - Sobhani, E. Improvement of Vibrational Characteristics of Multipurpose Structures (Plate and Shells) Used in Aerospace Components by Deploying Graphene Oxide Powders (GOPs) in a Matrix as a Nano-Reinforcement: A Comprehensive Study. Eng. Anal. Bound. Elem.
**2023**, 146, 598–635. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Ersoy, H.; Civalek, O. Solution of Moore–Gibson–Thompson Equation of an Unbounded Medium with a Cylindrical Hole. Mathematics
**2021**, 9, 1536. [Google Scholar] [CrossRef] - Miao, H.C.; Huan, Q.; Wang, Q.Z.; Li, F.X. Excitation and Reception of Single Torsional Wave T(0,1) Mode in Pipes Using Face-Shear d24 Piezoelectric Ring Array. Smart Mater. Struct.
**2017**, 26, 025021. [Google Scholar] [CrossRef] - Zhang, H.; Du, Y.H.; Tang, J.H.; Kang, G.Z.; Miao, H.C. Circumferential SH Wave Piezoelectric Transducer System for Monitoring Corrosion-Like Defect in Large-Diameter Pipes. Sensors
**2020**, 20, 460. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dabak, A. Data-Driven Semi-Supervised and Supervised Learning Algorithms for Health Monitoring of Pipes. Mech. Syst. Signal Process.
**2019**, 131, 524–537. [Google Scholar] - Yeung, C.; Ng, C.T. Time-Domain Spectral Finite Element Method for Analysis of Torsional Guided Waves Scattering and Mode Conversion by Cracks in Pipes. Mech. Syst. Signal Process.
**2019**, 128, 305–317. [Google Scholar] [CrossRef] - Wang, Z.; Wang, S.; Wang, Q.; Zhao, W.; Huang, S.L. Development of a Helical Lamb Wave Electromagnetic Acoustic Transducer for Pipeline Inspection. IEEE Sens. J.
**2020**, 20, 9715–9723. [Google Scholar] [CrossRef] - Guan, R.Q.; Lu, Y.; Wang, K.; Su, Z.Q. Fatigue Crack Detection in Pipes with Multiple Mode Nonlinear Guided Waves. Struct. Health Monit.-Int. J.
**2019**, 18, 180–192. [Google Scholar] [CrossRef] - El Najjar, J.; Mustapha, S. Condition Assessment of Timber Utility Poles Using Ultrasonic Guided Waves. Constr. Build. Mater.
**2020**, 272, 121902. [Google Scholar] [CrossRef] - Majhi, S.; Mukherjee, A.; George, N.V.; Uy, B. Corrosion Detection in Steel Bar: A Time-Frequency Approach. NDT E Int.
**2019**, 107, 102150. [Google Scholar] [CrossRef] - Sun, K.; Meng, G.; Li, F.; Ye, L.; Lu, Y. Damage Identification in Thick Steel Beam Based on Guided Ultrasonic Waves. J. Intell. Mater. Syst. Struct.
**2010**, 21, 225–232. [Google Scholar] [CrossRef] - Song, G.; Li, H.; Gajic, B.; Zhou, W.; Chen, P.; Gu, H. Wind Turbine Blade Health Monitoring with Piezoceramic-Based Wireless Sensor Network. Int. J. Smart Nano Mater.
**2013**, 4, 150–166. [Google Scholar] [CrossRef] - Rose, J.L.; Ditri, J.J.; Pilarski, A.; Rajana, K.; Carr, F. A Guided Wave Inspection Technique for Nuclear Steam Generator Tubing. NDT E Int.
**1994**, 27, 307–310. [Google Scholar] [CrossRef] - Shin, H.J.; Rose, J.L. Guided Waves by Axisymmetric and Non-Axisymmetric Surface Loading on Hollow Cylinders. Ultrasonics
**1999**, 37, 355–363. [Google Scholar] [CrossRef] [PubMed] - Geetha, G.K.; Gopalakrishnan, S.; Hanagud, S. Laser Doppler Imaging of Delamination in a Composite T-Joint with Remotely Located Ultrasonic Actuators. Compos. Struct.
**2016**, 147, 197–210. [Google Scholar] [CrossRef] - Serey, V.; Quaegebeur, N.; Renier, M.; Micheau, P.; Masson, P.; Castaings, M. Selective Generation of Ultrasonic Guided Waves for Damage Detection in Rectangular Bars. Struct. Health Monit.-Int. J.
**2020**, 20, 1156–1168. [Google Scholar] [CrossRef] - Zhang, J.Q.; Wu, Z.J.; Yang, Z.Y.; Liu, K.H.; Zhou, K.; Zheng, Y.B. Excitation of Guided Wave Modes in Arbitrary Cross-Section Structures by Applied Surface Tractions. Smart Mater. Struct.
**2020**, 29, 065010. [Google Scholar] [CrossRef] - Tu, J.Q.; Tang, Z.F.; Yun, C.B.; Wu, J.; Xu, X. Guided Wave-Based Damage Assessment on Welded Steel I-Beam Under Ambient Temperature Variations. Struct. Control Health Monit.
**2021**, 28, e2696. [Google Scholar] [CrossRef] - Ditri, J.J.; Rose, J.L. Excitation of Guided Elastic Wave Modes in Hollow Cylinders by Applied Surface Tractions. J. Appl. Phys.
**1992**, 72, 2589–2597. [Google Scholar] [CrossRef] - Drygala, I.J.; Dulinska, J.M. Full-Scale Experimental and Numerical Investigations on the Modal Parameters of a Single-Span Steel-Frame Footbridge. Symmetry
**2019**, 11, 404. [Google Scholar] [CrossRef] [Green Version] - Yang, Y.; Yang, L.; Yao, G. Post-Processing of High Formwork Monitoring Data Based on the Back Propagation Neural Networks Model and the Autoregressive-Moving-Average Model. Symmetry
**2021**, 13, 1543. [Google Scholar] [CrossRef] - Yao, G.; Sun, Y.J.; Wong, M.P.; Lv, X.N. A Real-Time Detection Method for Concrete Surface Cracks Based on Improved YOLOv4. Symmetry
**2021**, 13, 1716. [Google Scholar] [CrossRef] - Hayashi, T.; Rose, J.L. Guided Wave Simulation and Visualization by A Semi-Analytical Finite Element Method. Mater. Eval.
**2003**, 61, 75–79. [Google Scholar] - Hayashi, T.; Kawashima, K.; Sun, Z.Q.; Rose, J.L. Analysis of Flexural Mode Focusing by a Semi-Analytical Finite Element Method. J. Acoust. Soc. Am.
**2003**, 113, 1241–1248. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hayashi, T.; Song, W.J.; Rose, J.L. Guided Wave Dispersion Curves for a Bar with an Arbitrary Cross-Section, a Rod and Rail Example. Ultrasonics
**2004**, 41, 175–183. [Google Scholar] [CrossRef] [PubMed] - Xie, J.; Ding, W.; Zou, W.; Wang, T.; Yang, J. Defect Detection inside a Rail Head by Ultrasonic Guided Waves. Symmetry
**2022**, 14, 2566. [Google Scholar] [CrossRef]

**Figure 2.**Cross-section and mesh generation of square steel tube: (

**a**) cross-section (unit: mm); (

**b**) mesh generation.

**Figure 4.**Wave structure for various points of the section: (

**a**) the first mode; (

**b**) the second mode; (

**c**) the fifth mode; (

**d**) the sixth mode.

**Figure 5.**The propagation path of UGWs under unilateral excitation: (

**a**) without damage; (

**b**) with damage.

**Figure 6.**The propagation path of UGWs under symmetric excitation: (

**a**) without damage; (

**b**) with damage.

**Figure 7.**Finite element model of square steel tube: (

**a**) square steel tube model; (

**b**) cross-section meshing.

**Figure 10.**Displacement field of guided wave in long edge: (

**a**) 60 μs; (

**b**) 200 μs; (

**c**) 400 μs; (

**d**) 600 μs.

**Figure 11.**Signal received from the symmetrically arranged receiving sensor under symmetric excitation: (

**a**) received signal on one side; (

**b**) received signal of the symmetrical position.

**Figure 14.**Guided wave propagation in damaged square steel tube: (

**a**) 240 ms; (

**b**) 440 ms; (

**c**) 600 ms.

**Figure 18.**Undamaged signals from short edge: (

**a**) signal at one path; (

**b**) symmetrical position signal; (

**c**) difference signal.

**Figure 20.**Schematic diagram of round hole position: (

**a**) middle hole; (

**b**) offset hole; (

**c**) corner hole.

**Figure 23.**Hole damage identification on long edge: (

**a**) middle hole; (

**b**) offset hole; (

**c**) corner hole.

**Figure 24.**Hole damage identification on short edge: (

**a**) middle hole; (

**b**) offset hole; (

**c**) corner hole.

Case No. | Crack Length (cm) | Position | Tube Length (m) | Defect Location (m) |
---|---|---|---|---|

1 | 2 | Long edge | 2 | 1 |

2 | Short edge | 2 | 1 | |

3 | 3 | Long edge | 2 | 1 |

4 | Short edge | 2 | 1 | |

5 | 4 | Long edge | 2 | 1 |

6 | Short edge | 2 | 1 |

Case No. | Damage Location | Position | Tube Length (m) | Defect Location (m) |
---|---|---|---|---|

7 | Middle hole | Long edge | 3 | 1.5 |

8 | Short edge | 3 | 2 | |

9 | Offset hole | Long edge | 2 | 1.5 |

10 | Short edge | 2 | 1 | |

11 | Corner hole | Long edge | 2 | 1.5 |

12 | Short edge | 2 | 1 |

Case No. | Time Difference (s) | Tube Length (m) | Identified Defect Location (m) | Actual Defect Location (m) | Inspection Error |
---|---|---|---|---|---|

1 | 868 | 2 | 1.09 | 1 | 8.98% |

2 | 843 | 2 | 1.015 | 1 | 1.51% |

3 | 869 | 2 | 1.091 | 1 | 9.11% |

4 | 844 | 2 | 1.016 | 1 | 1.63% |

5 | 868 | 2 | 1.09 | 1 | 8.98% |

6 | 844 | 2 | 1.016 | 1 | 1.63% |

Case No. | Time Difference (s) | Tube Length (m) | Identified Defect Location (m) | Actual Defect Location (m) | Inspection Error |
---|---|---|---|---|---|

7 | 1232 | 3 | 1.543 | 1.5 | 2.84% |

8 | 1642 | 3 | 1.968 | 2 | 1.62% |

9 | 1258 | 2 | 1.575 | 1.5 | 5.00% |

10 | 837 | 2 | 1.008 | 1 | 0.79% |

11 | 1383 | 2 | 1.731 | 1.5 | 15.37% |

12 | 974 | 2 | 1.171 | 1 | 17.13% |

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**MDPI and ACS Style**

Yang, T.; Zhou, W.; Yu, L.
Guided Wave-Based Damage Detection of Square Steel Tubes Utilizing Structure Symmetry. *Symmetry* **2023**, *15*, 805.
https://doi.org/10.3390/sym15040805

**AMA Style**

Yang T, Zhou W, Yu L.
Guided Wave-Based Damage Detection of Square Steel Tubes Utilizing Structure Symmetry. *Symmetry*. 2023; 15(4):805.
https://doi.org/10.3390/sym15040805

**Chicago/Turabian Style**

Yang, Tingting, Wensong Zhou, and Lei Yu.
2023. "Guided Wave-Based Damage Detection of Square Steel Tubes Utilizing Structure Symmetry" *Symmetry* 15, no. 4: 805.
https://doi.org/10.3390/sym15040805