# Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Application of MFP Algorithms to Damage Identification

#### 2.1. Bartlett Beamformer

#### 2.2. Minimum Variance Distortionless Response (MVDR) Beamformer

## 3. Direct Problem

#### 3.1. Sensitivity of Modal Quantities to Local Damage and Generation of the Replica Vector

#### 3.2. Comparison between Numerical and Experimental Results

## 4. Inverse Problem

#### 4.1. Pseudo-Experimental Data

#### 4.1.1. Damage Located at a Sensor Point

#### 4.1.2. Damage Located in-between Sensors

#### 4.2. Experimental Tests

#### 4.2.1. Damage Located at a Sensor Point: Case A

#### 4.2.2. Damage Located in between Sensors: Case B

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Xiao, F.; Zhu, W.; Meng, X.; Chen, G.S. Parameter identification of frame structures by considering shear deformation. Int. J. Distrib. Sens. Netw.
**2023**, 2023, 6631716. [Google Scholar] [CrossRef] - Meng, X.; Xiao, F.; Yan, Y.; Ma, Y. Non-destructive damage evaluation based on static response for beam-like structures considering shear deformation. Appl. Sci.
**2023**, 13, 8219. [Google Scholar] [CrossRef] - Farrar, C.R.; Doebling, S.W. Damage detection II: Field applications to large structures. In Modal Analysis and Testing; Nato Science Series; Silva, J.M.M., Maia, N.M.M., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1999. [Google Scholar]
- Dimarogonas, A.D. Vibration of Cracked Structures: A State of the Art Review. Eng. Fract. Mech.
**1996**, 55, 831–857. [Google Scholar] [CrossRef] - Doebling, S.W.; Farrar, C.R.; Prime, M.B.; Shevitz, D.W. Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review; Los Alamos National Loboratory report LA-13070-MS; Los Alamos National Loboratory: Los Alamos, NM, USA, 1996. [Google Scholar]
- Pau, A.; Vestroni, F. Vibration analysis and dynamic characterization of the Colosseum. Struct. Control Health Monit.
**2008**, 15, 1105–1121. [Google Scholar] [CrossRef] - Fan, W.; Qiao, P. Vibration-based Damage Identification Methods: A Review and Comparative Study. Struct. Health Monit.
**2011**, 10, 83–111. [Google Scholar] [CrossRef] - Eroglu, U.; Tufekci, E. Exact solution based finite element formulation of cracked beams for crack detection. Int. J. Solids Struct.
**2016**, 96, 240–253. [Google Scholar] [CrossRef] - Deraemaeker, A.; Reynders, E.; De Roeck, G.; Kullaa, J. Vibration based SHM: Comparison of the performance of modal features vs features extracted from spatial filters under changing environmental conditions. In Proceedings of the ISMA2006 International Conference on Noise and Vibration Engineering, Leuven, Belgium, 18–20 September 2006; pp. 849–864. [Google Scholar]
- Hou, R.; Xia, Y. Review on the new development of vibration-based damage identification for civil engineering structures: 2010–2019. J. Sound Vib.
**2021**, 491, 115741. [Google Scholar] [CrossRef] - Abdel Wahab, M.; De Roeck, G. Damage detection in bridges using modal curvatures: Application to a real damage scenario. J. Sound Vib.
**1999**, 226, 217–235. [Google Scholar] [CrossRef] - Nguyen, D.H.; Nguyen, Q.B.; Bui-Tien, T.; De Roeck, G.; Abdel Wahab, M. Damage detection in girder bridges using modal curvatures gapped smoothing method and Convolutional Neural Network: Application to Bo Nghi bridge. Theor. Appl. Fract. Mech.
**2020**, 109, 102728. [Google Scholar] [CrossRef] - Dilena, M.; Morassi, A.; Perin, M. Dynamic identification of a reinforced concrete damaged bridge. Mech. Syst. Signal Process.
**2011**, 25, 2990–3009. [Google Scholar] [CrossRef] - Chandrashekhar, M.; Ganguli, R. Structural Damage Detection Using Modal Curvature and Fuzzy Logic. Struct. Health Monit.
**2009**, 8, 267–282. [Google Scholar] [CrossRef] - Pandey, A.; Biswas, M.; Samman, M. Damage detection from changes in curvature mode shapes. J. Sound Vib.
**1991**, 145, 321–332. [Google Scholar] [CrossRef] - De Roeck, G.; Reynders, E.; Anastasopoulos, D. Assessment of small damage by direct modal strain measurements. Lect. Notes Civ. Eng.
**2018**, 5, 3–16. [Google Scholar] - Cao, M.R.; Xu, W.; Radzieński, M.; Ostachowicz, W. Identification of multiple damage in beams based on robust curvature mode shapes. Mech. Syst. Signal Process.
**2014**, 46, 468–480. [Google Scholar] [CrossRef] - Ciambella, J.; Vestroni, F. The use of modal curvatures for damage localizationin beam-type structures. J. Sound Vib.
**2015**, 340, 126–137. [Google Scholar] [CrossRef] - Ciambella, J.; Pau, A.; Vestroni, F. Modal curvature-based damage localization in weakly damaged continuous beams. Mech. Syst. Signal Process.
**2019**, 121, 171–182. [Google Scholar] [CrossRef] - Vestroni, F.; Pau, A.; Ciambella, J. The role of curvatures in damage identification. In Proceedings of the Iabmas, Barcelona, Spain, 11–15 July 2022. [Google Scholar]
- Li, Y. Hypersensitivity of strain-based indicators for structural damage identification: A review. Mech. Syst. Signal Process.
**2010**, 24, 653–664. [Google Scholar] [CrossRef] - Dessi, D.; Camerlengo, G. Damage identification techniques via modal curvature analysis: Overview and comparison. Mech. Syst. Signal Process.
**2015**, 52–53, 181–205. [Google Scholar] [CrossRef] - Garrido, H.; Domizio, M.; Curadelli, O.; Ambrosini, D. Numerical, statistical and experimental investigation on damage quantification in beams from modal curvature. J. Sound Vib.
**2020**, 485, 115591. [Google Scholar] [CrossRef] - Capecchi, D.; Ciambella, J.; Pau, A.; Vestroni, F. Damage identification in a parabolic arch by means of natural frequencies, modal shapes and curvatures. Meccanica
**2016**, 51, 2847–2859. [Google Scholar] [CrossRef] - Eroglu, U.; Ruta, G.; Tufekci, E. Natural frequencies of parabolic arches with a single crack on opposite cross-section sides. J. Vib. Control
**2019**, 25, 1313–1325. [Google Scholar] [CrossRef] - Sternini, S.; Pau, A.; Di Scalea, F. Minimum-Variance Imaging in Plates Using Guided-Wave-Mode Beamforming. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2019**, 66, 1906–1919. [Google Scholar] [CrossRef] [PubMed] - Baggeroer, A.; Kuperman, W.; Mikhalevsky, P. An overview of matched field methods in ocean acoustics. IEEE J. Ocean. Eng.
**1993**, 18, 401–424. [Google Scholar] [CrossRef] - Chiariotti, P.; Martarelli, M.; Castellini, P. Acoustic beamforming for noise source localization—Reviews, methodology and applications. Mech. Syst. Signal Process.
**2019**, 120, 422–448. [Google Scholar] [CrossRef] - Turek, G.; Kuperman, W. Applications of matched-field processing to structural vibration problems. J. Acoust. Soc. Am.
**1997**, 101, 1430–1440. [Google Scholar] [CrossRef] - Tolstoy, A. Linearization of the matched field processing approach to acoustic tomography. J. Acoust. Soc. Am.
**1992**, 91, 781–787. [Google Scholar] [CrossRef] - Meng, W.; Ke, Y.; Li, J.; Zheng, C.; Li, X. Finite data performance analysis of one-bit MVDR and phase-only MVDR. Signal Process.
**2021**, 183, 108018. [Google Scholar] [CrossRef] - Jensen, F.B.; Kuperman, W.A.; Porter, M.B.; Schmidt, H. Computational Ocean Acoustics, 2nd ed.; Springer Publishing Company, Incorporated: New York, NY, USA, 2011. [Google Scholar]
- Dawari, V.; Vesmawala, G. Modal curvature and modal flexibility methods for honeycomb damage identification in reinforced concrete beams. Procedia Eng.
**2013**, 51, 119–124. [Google Scholar] [CrossRef] - Goyder, H. Methods and application of structural modelling from measured structural frequency response data. J. Sound Vib.
**1980**, 68, 209–230. [Google Scholar] [CrossRef] - Pau, A.; Greco, A.; Vestroni, F. Numerical and experimental detection of concentrated damage in a parabolic arch by measured frequency variations. J. Vib. Control
**2011**, 17, 605–614. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of the beam longitudinal view (

**left**) and of its cross section (

**right**).

**Figure 2.**Variation of modal displacements (

**a**–

**c**) and modal curvatures (

**d**–

**f**) in the four damage scenarios of location A, for the first (

**a**,

**d**), second (

**b**,

**e**) and third (

**c**,

**f**) mode.

**Figure 3.**Schematic representation of the experimental setup (

**a**) and image of the beam in laboratory conditions (

**b**).

**Figure 4.**Case A: experimental (e) and numerical (n) modal curvatures (first (

**a**), second (

**b**) and third (

**c**) mode) and their variations due to damage (first (

**d**), second (

**e**) and third (

**f**) mode).

**Figure 5.**Case B: experimental (e) and numerical (n) modal curvatures (first (

**a**), second (

**b**) and third (

**c**) mode) and their variations (first (

**d**), second (

**e**) and third (

**f**) mode) due to damage.

**Figure 6.**Contour plots of the objective function including one ((

**a**) i = 1, (

**b**) i = 2, (

**c**) i = 3) and three numerical modal curvatures ((

**d**) i = 1, 2, 3) (coordinates of the circle are ${s}^{D}$ and ${h}^{D}$ for D2.A).

**Figure 7.**Contour plots of the Bartlett beamformer including one ((

**a**) i = 1, (

**b**) i = 2, (

**c**) i = 3) and three numerical modal curvatures ((

**d**) i = 1, 2, 3) (coordinates of the circle are ${s}^{D}$ and ${h}^{D}$ for D2.A).

**Figure 8.**Contour plots of the MVDR beamformer including one ((

**a**) i = 1, (

**b**) i = 2, (

**c**) i = 3) and three numerical modal curvatures ((

**d**) i = 1, 2, 3) (coordinates of the circle are ${s}^{D}$ and ${h}^{D}$ for D2.A).

**Figure 9.**Contour plots of the objective function including one ((

**a**) i = 1, (

**b**) i = 2, (

**c**) i = 3) and three numerical modal curvatures ((

**d**) i = 1, 2, 3) (Circle: Correct damage parameters).

**Figure 10.**Contour plots of the Bartlett beamformer including one ((

**a**) i = 1, (

**b**) i = 2, (

**c**) i = 3) and three numerical modal curvatures ((

**d**) i = 1, 2, 3) (circle: correct damage parameters).

**Figure 11.**Contour plots of the MVDR beamformer including one ((

**a**) i = 1, (

**b**) i = 2, (

**c**) i = 3) and three numerical modal curvatures ((

**d**) i = 1, 2, 3) (circle: correct damage parameters).

**Figure 12.**Contour plots of the estimators for damage scenario D1.A ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (circle: correct damage parameters; cross: identified parameters).

**Figure 13.**Contour plots of the estimators for damage scenario D2.A ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (circle: correct damage parameters; cross: identified parameters).

**Figure 14.**Contour plots of the estimators for damage scenario D3.A ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (circle: correct damage parameters; cross: identified parameters).

**Figure 15.**Contour plots of the estimators for damage scenario D4.A ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (circle: correct damage parameters; cross: identified parameters).

**Figure 16.**Contour plots of the estimators for damage scenario D1.B ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (thick circle: correct damage parameters; cross: identified parameters).

**Figure 17.**Contour plots of the estimators for damage scenario D2.B ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (thick circle: correct damage parameters; thin concentric circles: identified parameters).

**Figure 18.**Contour plots of the estimators for damage scenario D3.B ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (thick circle: correct damage parameters; thin concentric circles: identified parameters).

**Figure 19.**Contour plots of the estimators for damage scenario D4.B ((

**a**) objective function, (

**b**) Bartlett, (

**c**) MVDR) (circle: correct damage parameters; cross: identified parameters).

**Table 1.**Labels of the damage scenarios with location (${s}^{D}$), nominal height of the damaged cross-section (${H}^{D}$), and percent stiffness reduction ($\beta $).

D1.A | D2.A | D3.A | D4.A | D1.B | D2.B | D3.B | D4.B | |
---|---|---|---|---|---|---|---|---|

${s}^{D}$ | 0.25 | 0.25 | 0.25 | 0.25 | 0.4375 | 0.4375 | 0.4375 | 0.4375 |

${h}^{D}$ | $0.875$ | $0.750$ | $0.625$ | $0.500$ | $0.875$ | $0.750$ | $0.625$ | $0.500$ |

$\beta $ | 33.0 | 57.8 | 75.6 | 87.5 | 33.0 | 57.8 | 75.6 | 87.5 |

**Table 2.**Numerical natural frequencies (${f}_{ni}$) [Hz] of the first three modes ($i=1,2,3$) and their percent variation ($\Delta {f}_{ni}$) due to damage.

U | D1.A | D2.A | D3.A | D4.A | |
---|---|---|---|---|---|

${f}_{n1}$ | 298.96 | 298.23 | 296.12 | 292.06 284.09 | |

$\Delta {f}_{n1}$ | 0.25 | 0.95 | 2.31 | 4.98 | |

${f}_{n2}$ | 817.12 | 811.67 | 796.67 | 769.99 | 725.52 |

$\Delta {f}_{n2}$ | 0.67 | 2.50 | 5.77 | 11.21 | |

${f}_{n3}$ | 1582.02 | 1572.86 | 1549.21 | 1512.15 | 1546.32 |

$\Delta {f}_{n3}$ | 0.58 | 2.07 | 4.23 | 7.39 | |

U | D1.B | D2.B | D3.B | D4.B | |

${f}_{n1}$ | 298.96 | 295.28 | 290.78 | 279.95 | 261.29 |

$\Delta {f}_{n1}$ | 1.23 | 2.73 | 6.36 | 12.60 | |

${f}_{n2}$ | 817.12 | 810.97 | 809.69 | 800.44 | 785.54 |

$\Delta {f}_{n2}$ | 0.75 | 0.91 | 2.04 | 3.87 | |

${f}_{n3}$ | 1582.02 | 1570.62 | 1568.89 | 1552.45 | 1525.23 |

$\Delta {f}_{n3}$ | 0.72 | 0.83 | 1.87 | 3.59 |

**Table 3.**Damage scenarios with measured heights of the damaged cross section (${H}^{D}$ [mm]), and percent stiffness reduction ($\beta $).

D1.A | D2.A | D3.A | D4.A | D1.B | D2.B | D3.B | D4.B | |
---|---|---|---|---|---|---|---|---|

${H}^{D}$ | 13.3 | 11.3 | 9.6 | 7.6 | 13.2 | 11.0 | 9.4 | 7.5 |

$\beta $ | 30.3 | 57.2 | 73.8 | 87.0 | 31.9 | 60.6 | 75.4 | 87.5 |

H [mm] | B [mm] | ${\mathit{S}}_{\mathit{D}}$ [mm] | E [GPa] | $\mathit{\rho}$ [kg/m${}^{3}$] | |
---|---|---|---|---|---|

A | 15.40 | 30.12 | 130.0 | 206.5 | 7504.5 |

B | 15.40 | 30.20 | 227.5 | 209.0 | 7656.8 |

**Table 5.**Case A: experimental (${f}_{ei}$) and numerical (${f}_{ni}$) natural frequencies [Hz] of the first three modes ($i=1,2,3$), and their percent variation ($\Delta {f}_{ei}$, $\Delta {f}_{ni}$) due to damage.

${f}_{e1}$ | $\Delta {f}_{e1}$ | ${f}_{e2}$ | $\Delta {f}_{e2}$ | ${f}_{e3}$ | $\Delta {f}_{e3}$ | |

U | 298.8 | 813.52 | 1589.8 | |||

D1.A | 298.2 | 0.21 | 810.5 | 0.38 | 1584.6 | 0.32 |

D2.A | 296.8 | 0.68 | 797.8 | 1.98 | 1561.3 | 1.79 |

D3.A | 294.1 | 1.58 | 784.5 | 3.57 | 1540.3 | 3.11 |

D4.A | 288.7 | 3.40 | 753.2 | 7.41 | 1497.6 | 5.80 |

${f}_{n1}$ | $\Delta {f}_{n1}$ | ${f}_{n2}$ | $\Delta {f}_{n2}$ | ${f}_{n3}$ | $\Delta {f}_{n3}$ | |

U | 298.8 | 814.5 | 1588.3 | |||

D1.A | 298.2 | 0.17 | 810.6 | 0.47 | 1581.8 | 0.41 |

D2.A | 296.5 | 0.75 | 798.1 | 2.00 | 1561.6 | 1.68 |

D3.A | 293.7 | 1.69 | 779.1 | 4.34 | 1533.5 | 3.45 |

D4.A | 288.0 | 3.61 | 744.3 | 8.62 | 1489.4 | 6.23 |

**Table 6.**Case B: experimental (${f}_{ei}$) and numerical (${f}_{ni}$) natural frequencies [Hz] of the first three modes ($i=1,2,3$), and their percent variation ($\Delta {f}_{ei}$, $\Delta {f}_{ni}$) due to damage.

${f}_{e1}$ | $\Delta {f}_{e1}$ | ${f}_{e2}$ | $\Delta {f}_{e2}$ | ${f}_{e3}$ | $\Delta {f}_{e3}$ | |

U | 297.7 | 809.8 | 1584.5 | |||

D1.B | 295.5 | 0.73 | 808.3 | 0.19 | 1579.0 | 0.35 |

D2.B | 287.8 | 3.33 | 803.5 | 0.78 | 1564.1 | 1.29 |

D3.B | 282.1 | 5.23 | 799.0 | 1.34 | 1549.6 | 2.20 |

D4.B | 267.6 | 10.12 | 790.0 | 2.45 | 1521.5 | 3.97 |

${f}_{n1}$ | $\Delta {f}_{n1}$ | ${f}_{n2}$ | $\Delta {f}_{n2}$ | ${f}_{n3}$ | $\Delta {f}_{n3}$ | |

U | 297.5 | 811.2 | 1582.6 | |||

D1.B | 295.1 | 0.83 | 808.9 | 0.28 | 1578.7 | 0.26 |

D2.B | 288.1 | 3.16 | 802.6 | 1.06 | 1567.5 | 0.96 |

D3.B | 279.1 | 6.19 | 794.9 | 2.01 | 1553.5 | 1.85 |

D4.B | 265.7 | 10.68 | 784.3 | 3.32 | 1534.2 | 3.06 |

D1.A | D2.A | D3.A | D4.A | |||||
---|---|---|---|---|---|---|---|---|

${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | ${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | ${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | ${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | |

H | 100 | −12.7 | 0.0 | 6.9 | 0.0 | 11.3 | 0.0 | 14.3 |

Bartlett | 0.0 | 5.8 | 0.0 | 6.9 | 0.0 | 11.3 | 0.0 | 14.3 |

MVDR | 0.0 | 5.8 | 0.0 | 6.9 | 0.0 | 11.3 | 0.0 | 14.3 |

D1.B | D2.B | D3.B | D4.B | |||||
---|---|---|---|---|---|---|---|---|

${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | ${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | ${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | ${\mathit{s}}^{\mathit{D}}$ | ${\mathit{h}}^{\mathit{D}}$ | |

H | 9.1 | −21.6 | 4.6 | −5.5 | 4.6 | 0.0 | 0.0 | 0.0 |

Bartlett | 9.1 | −18.2 | 4.6 | −1.4 | 0.0 | 0.0 | 0.0 | 0.0 |

MVDR | 9.1 | −8.0 | 4.6 | −1.4 | 0.0 | 0.0 | 0.0 | 0.0 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pau, A.; Eroğlu, U.
Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers. *Appl. Sci.* **2023**, *13*, 10557.
https://doi.org/10.3390/app131910557

**AMA Style**

Pau A, Eroğlu U.
Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers. *Applied Sciences*. 2023; 13(19):10557.
https://doi.org/10.3390/app131910557

**Chicago/Turabian Style**

Pau, Annamaria, and Uğurcan Eroğlu.
2023. "Identification of Damage in Beams by Modal Curvatures Using Acoustic Beamformers" *Applied Sciences* 13, no. 19: 10557.
https://doi.org/10.3390/app131910557