{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:57:14Z","timestamp":1774630634681,"version":"3.50.1"},"reference-count":63,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2022,10,30]],"date-time":"2022-10-30T00:00:00Z","timestamp":1667088000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Italian National Research Program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Axially loaded beam-like structures represent a challenging case study for unsupervised learning vibration-based damage detection. Under real environmental and operational conditions, changes in axial load cause changes in the characteristics of the dynamic response that are significantly greater than those due to damage at an early stage. In previous works, the authors proposed the adoption of a multivariate damage feature composed of eigenfrequencies of multiple vibration modes. Successful results were obtained by framing the problem of damage detection as that of unsupervised outlier detection, adopting the well-known Mahalanobis squared distance (MSD) to define an effective damage index. Starting from these promising results, a novel approach based on unsupervised learning data clustering is proposed in this work, which increases the sensitivity to damage and significantly reduces the uncertainty associated with the results, allowing for earlier damage detection. The novel approach, which is based on Gaussian mixture model, is compared with the benchmark one based on the MSD, under the effects of an uncontrolled environment and, most importantly, in the presence of real damage due to corrosion.<\/jats:p>","DOI":"10.3390\/s22218336","type":"journal-article","created":{"date-parts":[[2022,10,30]],"date-time":"2022-10-30T10:47:57Z","timestamp":1667126877000},"page":"8336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["A Damage Detection Approach for Axially Loaded Beam-like Structures Based on Gaussian Mixture Model"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6390-3704","authenticated-orcid":false,"given":"Francescantonio","family":"Luc\u00e0","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Politecnico di Milano, Via La Masa, 1-20156 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9240-5472","authenticated-orcid":false,"given":"Stefano","family":"Manzoni","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Politecnico di Milano, Via La Masa, 1-20156 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Cerutti","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Politecnico di Milano, Via La Masa, 1-20156 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6861-8374","authenticated-orcid":false,"given":"Alfredo","family":"Cigada","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Politecnico di Milano, Via La Masa, 1-20156 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1098\/rsta.2006.1928","article-title":"An introduction to structural health monitoring","volume":"365","author":"Farrar","year":"2007","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Farrar, C.R., and Worden, K. (2012). Structural Health Monitoring: A Machine Learning Perspective, John Wiley and Sons.","DOI":"10.1002\/9781118443118"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.ndteint.2017.11.004","article-title":"Design of an instrumentation for the automated damage detection in ceilings","volume":"94","author":"Belletti","year":"2018","journal-title":"NDT E Int."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Grosse, C.U., Gehlen, C., and Glaser, S.D. (2007). Sensing methods in civil engineering for an efficient construction management. Advances in Construction Materials 2007, Springer.","DOI":"10.1007\/978-3-540-72448-3_56"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1098\/rsta.2006.1935","article-title":"Effects of environmental and operational variability on structural health monitoring","volume":"365","author":"Sohn","year":"2007","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1098\/rsta.2000.0717","article-title":"Vibration-based structural damage identification","volume":"359","author":"Farrar","year":"2001","journal-title":"Philos. Trans. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1080\/10589759.2019.1649400","article-title":"Detection of cracks in axially loaded tie-rods by vibration analysis","volume":"35","author":"Collini","year":"2020","journal-title":"Nondestruct. Test. Eval."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1080\/15583058.2018.1563231","article-title":"Dynamical Assessment of the Work Conditions of Reinforcement Tie-Rods in Historical Masonry Structures","volume":"13","author":"Collini","year":"2019","journal-title":"Int. J. Archit. Herit."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.mechrescom.2012.06.005","article-title":"Bending tests to estimate the axial force in tie-rods","volume":"44","author":"Tullini","year":"2012","journal-title":"Mech. Res. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1061\/(ASCE)0733-9399(2001)127:12(1275)","article-title":"Experimental Methods for Estimating In Situ Tensile Force in Tie-Rods","volume":"127","author":"Bati","year":"2001","journal-title":"J. Eng. Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"241","DOI":"10.2749\/101686694780601809","article-title":"Determining the Axial Force in Metallic Rods","volume":"4","author":"Blasi","year":"1994","journal-title":"Struct. Eng. Int."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"111568","DOI":"10.1016\/j.engstruct.2020.111568","article-title":"Sensitivity analysis of frequency-based tie-rod axial load evaluation methods","volume":"229","author":"Resta","year":"2021","journal-title":"Eng. Struct."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Resta, C., Chellini, G., and Falco, A.D. (2020). Dynamic assessment of axial load in tie-rods by means of acoustic measurements. Buildings, 10.","DOI":"10.3390\/buildings10020023"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/j.ymssp.2015.01.019","article-title":"Development and validation of an automated operational modal analysis algorithm for vibration-based monitoring and tensile load estimation","volume":"60","author":"Rainieri","year":"2015","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1016\/j.jsv.2018.02.062","article-title":"Dynamic identification of axial force and boundary restraints in tie rods and cables with uncertainty quantification using Set Inversion Via Interval Analysis","volume":"423","author":"Kernicky","year":"2018","journal-title":"J. Sound Vib."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4122","DOI":"10.1016\/j.jsv.2013.03.018","article-title":"Estimate of the axial force in slender beams with unknown boundary conditions using one flexural mode shape","volume":"332","author":"Rebecchi","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.jsv.2008.03.061","article-title":"Dynamic identification of beam axial loads using one flexural mode shape","volume":"318","author":"Tullini","year":"2008","journal-title":"J. Sound Vib."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1016\/j.engstruct.2005.01.008","article-title":"The dynamical identification of the tensile force in ancient tie-rods","volume":"27","author":"Lagomarsino","year":"2005","journal-title":"Eng. Struct."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Collini, L., Garziera, R., and Riabova, K. (2017). Vibration Analysis for Monitoring of Ancient Tie-Rods. Shock Vib., 2017.","DOI":"10.1155\/2017\/7591749"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Campagnari, S., Di Matteo, F., Manzoni, S., Scaccabarozzi, M., and Vanali, M. (2017). Estimation of axial load in tie-rods using experimental and operational modal analysis. J. Vib. Acoust. Trans. ASME, 139.","DOI":"10.1115\/1.4036108"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.jsv.2012.08.009","article-title":"Nondestructive characterization of tie-rods by means of dynamic testing, added masses and genetic algorithms","volume":"332","author":"Gentilini","year":"2013","journal-title":"J. Sound Vib."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"108547","DOI":"10.1016\/j.ymssp.2021.108547","article-title":"A vibration-based approach for health monitoring of tie-rods under uncertain environmental conditions","volume":"167","author":"Manzoni","year":"2022","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Luc\u00e0, F., Manzoni, S., Cigada, A., Barella, S., Gruttadauria, A., and Cerutti, F. (2022). Automatic Detection of Real Damage in Operating Tie-Rods. Sensors, 22.","DOI":"10.3390\/s22041370"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/0167-6393(95)00009-D","article-title":"Speaker identification and verification using Gaussian mixture speaker models","volume":"17","author":"Reynolds","year":"1995","journal-title":"Speech Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1006\/dspr.1999.0361","article-title":"Speaker verification using adapted Gaussian mixture models","volume":"10","author":"Reynolds","year":"2000","journal-title":"Digit. Signal Process. A Rev. J."},{"key":"ref_26","first-page":"246","article-title":"Adaptive background mixture models for real-time tracking","volume":"2","author":"Stauffer","year":"1999","journal-title":"Proc. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1109\/TPAMI.2010.223","article-title":"Robust point set registration using Gaussian mixture models","volume":"33","author":"Jian","year":"2011","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1722","DOI":"10.1177\/1045389X13484101","article-title":"A probabilistic approach for damage identification and crack mode classification in reinforced concrete structures","volume":"24","author":"Farhidzadeh","year":"2013","journal-title":"J. Intell. Mater. Syst. Struct."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.apacoust.2016.08.006","article-title":"Acoustic emission monitoring of reinforced concrete beams subjected to four-point-bending","volume":"117","author":"Prem","year":"2017","journal-title":"Appl. Acoust."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.cemconres.2019.03.001","article-title":"Machine learning based crack mode classification from unlabeled acoustic emission waveform features","volume":"121","author":"Das","year":"2019","journal-title":"Cem. Concr. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.compositesb.2018.02.028","article-title":"Clustering of interlaminar and intralaminar damages in laminated composites under indentation loading using Acoustic Emission","volume":"144","author":"Saeedifar","year":"2018","journal-title":"Compos. Part B Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"125001","DOI":"10.1088\/0964-1726\/23\/12\/125001","article-title":"On-line updating Gaussian mixture model for aircraft wing spar damage evaluation under time-varying boundary condition","volume":"23","author":"Qiu","year":"2014","journal-title":"Smart Mater. Struct."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.engstruct.2017.09.063","article-title":"Condition assessment of cables by pattern recognition of vehicle-induced cable tension ratio","volume":"155","author":"Li","year":"2018","journal-title":"Eng. Struct."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1115\/1.2718241","article-title":"Time series based structural damage detection algorithm using Gaussian Mixtures Modeling","volume":"129","author":"Nair","year":"2007","journal-title":"J. Dyn. Syst. Meas. Control. Trans. ASME"},{"key":"ref_35","first-page":"125001","article-title":"A Benchmark Problem for Structural Health Monitoring and Damage Detection","volume":"23","author":"Johnson","year":"2001","journal-title":"Smart Mater. Struct."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s13349-013-0038-3","article-title":"Linear approaches to modeling nonlinearities in long-term monitoring of bridges","volume":"3","author":"Figueiredo","year":"2013","journal-title":"J. Civ. Struct. Health Monit."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/978-3-030-81716-9_14","article-title":"Vibration-Based Damage Feature for Long-Term Structural Health Monitoring Under Realistic Environmental and Operational Variability","volume":"21","author":"Manzoni","year":"2022","journal-title":"Struct. Integr."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Luc\u00e0, F., Manzoni, S., and Cigada, A. (2022). Data Driven Damage Detection Strategy Under Uncontrolled Environment. Lecture Notes in Civil Engineering, Springer.","DOI":"10.1007\/978-3-031-07258-1_77"},{"key":"ref_39","unstructured":"Ewins, D.J. (2001). Modal Testing: Theory, Practice and Application, Wiley."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Brandt, A. (2011). Noise and Vibration Analysis: Signal Analysis and Experimental Procedures, Wiley.","DOI":"10.1002\/9780470978160"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/j.ymssp.2018.06.009","article-title":"A signal processing framework for operational modal analysis in time and frequency domain","volume":"115","author":"Brandt","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/j.ijmecsci.2019.02.014","article-title":"Closed-form equation for natural frequencies of beams under full range of axial loads modeled with a spring-mass system","volume":"153\u2013154","author":"Valle","year":"2019","journal-title":"Int. J. Mech. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1121\/1.1911144","article-title":"Bending Frequencies of Compressed Beams","volume":"44","author":"Galef","year":"1968","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/34.824819","article-title":"Statistical pattern recognition: A review","volume":"22","author":"Jain","year":"2000","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Chen, H.P., and Ni, Y.Q. (2018). Vibration-Based Damage Identification Methods. Structural Health Monitoring of Large Civil Engineering Structures, John Wiley and Sons.","DOI":"10.1002\/9781119166641.ch7"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.jsv.2017.06.033","article-title":"Scaling of mode shapes from operational modal analysis using harmonic forces","volume":"407","author":"Brandt","year":"2017","journal-title":"J. Sound Vib."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1709","DOI":"10.1007\/s12206-022-0307-3","article-title":"Modal strain energy-based updating procedure for damage detection: A numerical investigation","volume":"36","author":"Nguyen","year":"2022","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"115741","DOI":"10.1016\/j.jsv.2020.115741","article-title":"Review on the new development of vibration-based damage identification for civil engineering structures: 2010\u20132019","volume":"491","author":"Hou","year":"2021","journal-title":"J. Sound Vib."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Cao, M.S., Sha, G.G., Gao, Y.F., and Ostachowicz, W. (2017). Structural damage identification using damping: A compendium of uses and features. Smart Mater. Struct., 26.","DOI":"10.1088\/1361-665X\/aa550a"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Murtagh, P.J., Basu, B., and Broderick, B.M. (2005, January 24\u201328). Identification of modal damping ratios for a simplified wind turbine tower using fourier analysis. Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference\u2014DETC2005, Long Beach, CA, USA.","DOI":"10.1115\/DETC2005-85523"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1177\/1475921710365419","article-title":"Vibration-based damage identification methods: A review and comparative study","volume":"10","author":"Fan","year":"2011","journal-title":"Struct. Health Monit."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1002\/1096-9845(200102)30:2<149::AID-EQE1>3.0.CO;2-Z","article-title":"One-year monitoring of the Z24Bridge: Environmental effects versus damage events","volume":"30","author":"Peeters","year":"2015","journal-title":"Earthq. Eng. & Struct. Dyn."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"647","DOI":"10.1006\/jsvi.1999.2514","article-title":"Damage detection using outlier analysis","volume":"229","author":"Worden","year":"2000","journal-title":"J. Sound Vib."},{"key":"ref_54","first-page":"1","article-title":"Machine learning algorithms for damage detection","volume":"1908","author":"Figueiredo","year":"2018","journal-title":"Vib.-Based Tech. Damage Detect. Localization Eng. Struct."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"106495","DOI":"10.1016\/j.ymssp.2019.106495","article-title":"A novel anomaly detection method based on adaptive Mahalanobis-squared distance and one-class kNN rule for structural health monitoring under environmental effects","volume":"140","author":"Sarmadi","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.ymssp.2011.06.009","article-title":"Identifying damage locations under ambient vibrations utilizing vector autoregressive models and Mahalanobis distances","volume":"26","author":"Mosavi","year":"2012","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ymssp.2017.11.045","article-title":"A comparison of linear approaches to filter out environmental effects in structural health monitoring","volume":"105","author":"Deraemaeker","year":"2018","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_58","first-page":"543","article-title":"Data Clustering","volume":"31","author":"Wagstaff","year":"2012","journal-title":"Adv. Mach. Learn. Data Min. Astron."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"147592172210752","DOI":"10.1177\/14759217221075241","article-title":"Three decades of statistical pattern recognition paradigm for SHM of bridges","volume":"21","author":"Figueiredo","year":"2022","journal-title":"Struct. Health Monit."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum Likelihood from Incomplete Data Via the EM Algorithm","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc. Ser. Methodol."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1137\/S1052623496303470","article-title":"Convergence properties of the Nelder-Mead simplex method in low dimensions","volume":"9","author":"Lagarias","year":"1998","journal-title":"SIAM J. Optim."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1162\/neco.1996.8.1.129","article-title":"On Convergence Properties of the EM Algorithm for Gaussian Mixtures","volume":"8","author":"Xu","year":"1996","journal-title":"Neural Comput."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1016\/j.patrec.2009.09.011","article-title":"Data clustering: 50 years beyond K-means","volume":"31","author":"Jain","year":"2010","journal-title":"Pattern Recognit. Lett."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/21\/8336\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:06:07Z","timestamp":1760144767000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/21\/8336"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,30]]},"references-count":63,"journal-issue":{"issue":"21","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["s22218336"],"URL":"https:\/\/doi.org\/10.3390\/s22218336","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,30]]}}}