{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T04:48:45Z","timestamp":1768452525826,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,2,3]],"date-time":"2021-02-03T00:00:00Z","timestamp":1612310400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51678032"],"award-info":[{"award-number":["51678032"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper aims to present a method for quantitative damage identification of a simply supported beam, which integrates the frequency response function (FRF) and model updating. The objective function is established using the cross-signature assurance criterion (CSAC) indices of the FRFs between the measurement points and the natural frequency. The CSAC index in the frequency range between the first two frequencies is found to be sensitive to damage. The proposed identification procedure is tried to identify the single and multiple damages. To verify the effectiveness of the method, numerical simulation and laboratory testing were conducted on some model steel beams with simulated damage by cross-cut sections, and the identification results were compared with the real ones. The analysis results show that the proposed damage evaluation method is insensitive to the systematic test errors and is able to locate and quantify the damage within the beam structures step by step.<\/jats:p>","DOI":"10.3390\/s21041029","type":"journal-article","created":{"date-parts":[[2021,2,3]],"date-time":"2021-02-03T20:31:51Z","timestamp":1612384311000},"page":"1029","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Step-by-Step Damage Identification Method Based on Frequency Response Function and Cross Signature Assurance Criterion"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9346-2447","authenticated-orcid":false,"given":"Jiawang","family":"Zhan","sequence":"first","affiliation":[{"name":"School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9586-5848","authenticated-orcid":false,"given":"Fei","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4740-9503","authenticated-orcid":false,"given":"Mohammad","family":"Siahkouhi","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/j.engstruct.2019.02.037","article-title":"Method and monitoring approach for distributed detection of damage in multi-span continuous bridges","volume":"189","author":"Oskoui","year":"2019","journal-title":"Eng. Struct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"115088","DOI":"10.1016\/j.jsv.2019.115088","article-title":"Using drive-by health monitoring to detect bridge damage considering environmental and operational effects","volume":"468","author":"Locke","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107360","DOI":"10.1016\/j.measurement.2019.107360","article-title":"Comprehensive measurement techniques and multi-index correlative evaluation approach for structural health monitoring of highway bridges","volume":"152","author":"Zhou","year":"2020","journal-title":"Measurement"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Zhu, X., Cao, M., Ostachowicz, W., and Xu, W. (2019). Damage Identification in Bridges by Processing Dynamic Responses to Moving Loads: Features and Evaluation. Sensors, 19.","DOI":"10.3390\/s19030463"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1016\/j.engstruct.2016.11.035","article-title":"Vibration-based damage detection for structural connections using incomplete modal data by Bayesian approach and model reduction technique","volume":"132","author":"Yin","year":"2017","journal-title":"Eng. Struct."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.jsv.2016.04.025","article-title":"Vibration characteristics and damage detection in a suspension bridge","volume":"375","author":"Wickramasinghe","year":"2016","journal-title":"J. Sound Vib."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.csefa.2015.08.001","article-title":"Vibration measurement-based simple technique for damage detection of truss bridges: A case study","volume":"4","author":"Siriwardane","year":"2015","journal-title":"Case Stud. Eng. Fail. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1002\/stc.1838","article-title":"Mode shape-based damage detection in plate structure without baseline data","volume":"23","author":"Rucevskis","year":"2016","journal-title":"Struct. Control Health Monit."},{"key":"ref_9","first-page":"1","article-title":"Modal Strain Energy-based Structural Damage Identification: A Review and Comparative Study","volume":"29","author":"Wang","year":"2018","journal-title":"Struct. Eng. Int."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mondal, S., Mondal, B., Bhutia, A., and Chakraborty, S. (2015). Damage Detection in Beams Using Frequency Response Function Curvatures near resonating frequencies. Advances in Structural Engineering, Springer.","DOI":"10.1007\/978-81-322-2193-7_119"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Avci, O., Abdeljaber, O., Kiranyaz, S., Hussein, M., Gabbouj, M., and Inman, D. (2020). A Review of Vibration-Based Damage Detection in Civil Structures: From Traditional Methods to Machine Learning and Deep Learning Applications. arXiv.","DOI":"10.1016\/j.ymssp.2020.107077"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1061\/(ASCE)0733-9445(2002)128:1(96)","article-title":"Structural Damage Identification using Modal Data. II: Test Verification","volume":"128","author":"Ren","year":"2002","journal-title":"J. Struct. Eng."},{"key":"ref_13","unstructured":"Peeters, B. (2000). System Identification and Damage Detection in Civil Engineering. [Ph.D. Thesis, KU Leuven]."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Mousavi, A.A., Zhang, C., Masri, S.F., and Gholipour, G. (2020). Structural Damage Localization and Quantification Based on a CEEMDAN Hilbert Transform Neural Network Approach: A Model Steel Truss Bridge Case Study. Sensors, 20.","DOI":"10.3390\/s20051271"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"110891","DOI":"10.1016\/j.engstruct.2020.110891","article-title":"Drive-by-bridge inspection for damage identification in a cable-stayed bridge: Numerical investigations","volume":"223","author":"Kildashti","year":"2020","journal-title":"Eng. Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"111551","DOI":"10.1016\/j.engstruct.2020.111551","article-title":"A damage identification method for connections of adjacent box-beam bridges using vehicle-bridge interaction analysis and model updating","volume":"228","author":"Zhan","year":"2021","journal-title":"Eng. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2198","DOI":"10.1016\/j.ymssp.2006.10.002","article-title":"Development in vibration-based structural damage detection technique","volume":"21","author":"Yan","year":"2007","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1177\/1475921704047500","article-title":"Vibration Based Condition Monitoring: A Review","volume":"3","author":"Carden","year":"2004","journal-title":"Struct. Health Monit."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1061\/(ASCE)0733-9399(2002)128:5(521)","article-title":"Improved Damage Quantification from Elemental Modal Strain Energy Change","volume":"128","author":"Shi","year":"2002","journal-title":"J. Eng. Mech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2221","DOI":"10.1016\/j.engstruct.2010.03.024","article-title":"Structural damage detection from coupling forces between substructures under support excitation","volume":"32","author":"Law","year":"2010","journal-title":"Eng. Struct."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1624","DOI":"10.2514\/1.30191","article-title":"Structural Damage Detection Based on Proper Orthogonal Decomposition: Experimental Verification","volume":"46","author":"Galvanetto","year":"2008","journal-title":"AIAA J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1016\/j.jsv.2009.07.001","article-title":"Structural model updating using frequency response function and quasi-linear sensitivity equation","volume":"326","author":"Esfandiari","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1061\/(ASCE)0733-9445(2002)128:1(87)","article-title":"Structural Damage Identification using Modal Data. I: Simulation Verification","volume":"128","author":"Ren","year":"2002","journal-title":"J. Struct. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1002\/stc.1768","article-title":"A frequency response-based structural damage identification using model updating method","volume":"23","author":"Shadan","year":"2016","journal-title":"Struct. Control Health Monit."},{"key":"ref_25","unstructured":"Dascotte, A.E., and Strobbe, J. (1999, January 8\u201311). Updating Finite Element Models Using FRF Correlation Functions. Proceedings of the IMAC XVII: 17th International Modal Analysis Conference, Kissimmee, FL, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2200","DOI":"10.1016\/j.ymssp.2006.05.008","article-title":"Model updating of damped structures using FRF data","volume":"20","author":"Lin","year":"2006","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1965","DOI":"10.2514\/3.49114","article-title":"Design Sensitivity Analysis of Structural Frequency Response","volume":"31","author":"Ting","year":"1993","journal-title":"AIAA J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/j.ymssp.2015.09.036","article-title":"Model updating using correlation analysis of strain frequency response function","volume":"70","author":"Guo","year":"2016","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3657","DOI":"10.1016\/j.jsv.2014.03.015","article-title":"Structural model updating using incomplete transfer function of strain data","volume":"333","author":"Esfandiari","year":"2014","journal-title":"J. Sound Vib."},{"key":"ref_30","first-page":"784","article-title":"Finite element model updating using frequency response functions and numerical sensitivities","volume":"21","author":"Sipple","year":"2014","journal-title":"Struct. Control Health Monit."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1061\/(ASCE)0733-9399(2009)135:4(243)","article-title":"Bayesian Model Updating Using Hybrid Monte Carlo Simulation with Application to Structural Dynamic Models with Many Uncertain Parameters","volume":"135","author":"Cheung","year":"2009","journal-title":"J. Eng. Mech."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1006\/mssp.1995.0015","article-title":"Finite element model updating using frequency response function data: I. Theory and initial investigation","volume":"9","author":"Imregun","year":"1995","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1006\/jsvi.1999.2448","article-title":"Parameterization of damage in reinforced concrete structures using model updating","volume":"228","author":"Wahab","year":"1999","journal-title":"J. Sound Vib."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Yin, T., and Zhu, H.P. (2018). Probabilistic Damage Detection of a Steel Truss Bridge Model by Optimally Designed Bayesian Neural Network. Sensors, 18.","DOI":"10.3390\/s18103371"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Zhang, Z., and Sun, C. (2020). Structural damage identification via physics-guided machine learning: A methodology integrating pattern recognition with finite element model updating. Struct. Health Monit.","DOI":"10.1177\/1475921720927488"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"107473","DOI":"10.1016\/j.measurement.2020.107473","article-title":"Multi-site structural damage identification using a multi-label classification scheme of machine learning","volume":"154","author":"Zhang","year":"2020","journal-title":"Measurement"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1016\/j.jsv.2016.10.043","article-title":"Real-time vibration-based structural damage detection using one-dimensional convolutional neural networks","volume":"388","author":"Abdeljaber","year":"2017","journal-title":"J. Sound Vib."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"04015081","DOI":"10.1061\/(ASCE)BE.1943-5592.0000855","article-title":"Model Updating of a Concrete Beam with Extensive Distributed Damage Using Experimental Frequency Response Function","volume":"21","author":"Esfandiari","year":"2016","journal-title":"J. Bridge Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1061\/(ASCE)BE.1943-5592.0000228","article-title":"Instrumentation, Nondestructive Testing, and Finite-Element Model Updating for Bridge Evaluation Using Strain Measurements","volume":"17","author":"Sanayei","year":"2012","journal-title":"J. Bridge Eng."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/4\/1029\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:19:24Z","timestamp":1760159964000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/4\/1029"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,3]]},"references-count":39,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["s21041029"],"URL":"https:\/\/doi.org\/10.3390\/s21041029","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,3]]}}}