{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T05:14:15Z","timestamp":1773292455246,"version":"3.50.1"},"reference-count":52,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,29]],"date-time":"2021-12-29T00:00:00Z","timestamp":1640736000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Key R&amp;D Project of Jiangxi Science and Technology Department","award":["No. 20192BBH80021"],"award-info":[{"award-number":["No. 20192BBH80021"]}]},{"name":"the Nanjing Science and Technology Project","award":["No. 202002014"],"award-info":[{"award-number":["No. 202002014"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Curved beam bridges, whose line type is flexible and beautiful, are an indispensable bridge type in modern traffic engineering. Nevertheless, compared with linear bridges, curved beam bridges have more complex internal forces and deformation due to the curvature; therefore, this type of bridge is more likely to suffer damage in strong earthquakes. The occurrence of damage reduces the safety of bridges, and can even cause casualties and property loss. For this reason, it is of great significance to study the identification of seismic damage in curved beam bridges. However, there is currently little research on curved beam bridges. For this reason, this paper proposes a damage identification method based on wavelet packet norm entropy (WPNE) under seismic excitation. In this method, wavelet packet transform is adopted to highlight the damage singularity information, the Lp norm entropy of wavelet coefficient is taken as a damage characteristic factor, and then the occurrence of damage is characterized by changes in the damage index. To verify the feasibility and effectiveness of this method, a finite element model of Curved Continuous Rigid-Frame Bridges (CCRFB) is established for the purposes of numerical simulation. The results show that the damage index based on WPNE can accurately identify the damage location and characterize the severity of damage; moreover, WPNE is more capable of performing damage location and providing early warning than the method based on wavelet packet energy. In addition, noise resistance analysis shows that WPNE is immune to noise interference to a certain extent. As long as a series of frequency bands with larger correlation coefficients are selected for WPNE calculation, independent noise reduction can be achieved.<\/jats:p>","DOI":"10.3390\/s22010239","type":"journal-article","created":{"date-parts":[[2021,12,29]],"date-time":"2021-12-29T23:31:35Z","timestamp":1640820695000},"page":"239","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Seismic Damage Identification Method for Curved Beam Bridges Based on Wavelet Packet Norm Entropy"],"prefix":"10.3390","volume":"22","author":[{"given":"Tongfa","family":"Deng","sequence":"first","affiliation":[{"name":"Jiangxi Province Key Laboratory of Environmental Geotechnical Engineering and Hazards Control, Jiangxi University of Science and Technology, Ganzhou 341000, China"},{"name":"School of Civil and Surveying & Mapping Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinwen","family":"Huang","sequence":"additional","affiliation":[{"name":"Jiangxi Province Key Laboratory of Environmental Geotechnical Engineering and Hazards Control, Jiangxi University of Science and Technology, Ganzhou 341000, China"},{"name":"School of Civil and Surveying & Mapping Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Maosen","family":"Cao","sequence":"additional","affiliation":[{"name":"Jiangxi Province Key Laboratory of Environmental Geotechnical Engineering and Hazards Control, Jiangxi University of Science and Technology, Ganzhou 341000, China"},{"name":"Department of Engineering Mechanics, Hohai University, Nanjing 210098, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dayang","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Engineering Mechanics, Hohai University, Nanjing 210098, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mahmoud","family":"Bayat","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"112216","DOI":"10.1016\/j.engstruct.2021.112216","article-title":"Shake table experimental study of curved bridges with consideration of girder-to-girder collision","volume":"237","author":"Jiao","year":"2021","journal-title":"Eng. Struct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"04019040","DOI":"10.1061\/(ASCE)BE.1943-5592.0001403","article-title":"Design of Horizontally Curved Steel Bridge Girders with Tubular Flanges","volume":"24","author":"Ma","year":"2019","journal-title":"J. Bridge Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.engstruct.2011.09.008","article-title":"Horizontally curved steel bridge seismic vulnerability assessment","volume":"34","author":"Seo","year":"2012","journal-title":"Eng. Struct."},{"key":"ref_4","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_5","doi-asserted-by":"crossref","unstructured":"Liu, X., Jin, B.Q., Bai, Q., Wang, Y., Wang, D., and Wang, Y.C. (2016). Distributed Fiber-Optic Sensors for Vibration Detection. Sensors, 16.","DOI":"10.3390\/s16081164"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1177\/1475921719880980","article-title":"Damage identification from static tests by eigenparameter decomposition and sparse regularization","volume":"19","author":"Lu","year":"2020","journal-title":"Struct. Health Monit."},{"key":"ref_7","first-page":"1","article-title":"Experimental verification in static tests of a computational method for detecting the position of a defect","volume":"15","author":"Vito","year":"2019","journal-title":"Eur. J. Environ. Civ. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1080\/15376494.2018.1513612","article-title":"Static data based damage localization of beam-column structures considering axial load","volume":"27","author":"Yazdanpanah","year":"2020","journal-title":"Mech. Adv. Mater. Struc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1080\/17415977.2018.1553965","article-title":"Static damage identification in beams by minimum constitutive relation error","volume":"27","author":"Guo","year":"2019","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/s13349-020-00394-4","article-title":"Seismic damage identification by fitting the nonlinear and hysteretic dynamic response of monitored buildings","volume":"10","author":"Miraglia","year":"2020","journal-title":"J. Civ. Struct. Health"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"106444","DOI":"10.1016\/j.ymssp.2019.106444","article-title":"Bayesian probabilistic damage characterization based on a perturbation model using responses at vibration nodes","volume":"139","author":"Huang","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1007\/s40999-020-00508-5","article-title":"Progressive Damage Analyses of Masonry Buildings by Dynamic Analyses","volume":"18","author":"Aras","year":"2020","journal-title":"Int. J. Civ. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1177\/1475921720932614","article-title":"Deep neural networks-based damage detection using vibration signals of finite element model and real intact state: An evaluation via a lab-scale offshore jacket structure","volume":"20","author":"Mousavi","year":"2021","journal-title":"Struct. Health Monit."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2477","DOI":"10.1007\/s00158-019-02336-1","article-title":"A piecewise constant level set method for damage identification of continuum structures based on natural frequencies","volume":"60","author":"Zhang","year":"2019","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"103030","DOI":"10.1016\/j.probengmech.2020.103030","article-title":"A compressive MUSIC spectral approach for identification of closely-spaced structural natural frequencies and post-earthquake damage detection","volume":"60","author":"Gkoktsi","year":"2020","journal-title":"Probabilist. Eng. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.compstruct.2018.02.028","article-title":"Damage identification method based on continuous wavelet transform and mode shapes for composite laminates with cutouts","volume":"191","author":"Zhou","year":"2018","journal-title":"Compos. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jsv.2019.06.011","article-title":"The efficiency of a novel identification method for structural damage assessment using the first vibration mode data","volume":"458","author":"Jahangiri","year":"2019","journal-title":"J. Sound Vib."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.ymssp.2018.11.012","article-title":"Modal curvature-based damage localization in weakly damaged continuous beams","volume":"121","author":"Ciambella","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.compstruct.2016.04.001","article-title":"Two-dimensional modal curvature estimation via Fourier spectral method for damage detection","volume":"148","author":"Yang","year":"2016","journal-title":"Compos. Struct."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"104228","DOI":"10.1016\/j.engfailanal.2019.104228","article-title":"Damage detection in wind turbine gearbox using modal strain energy","volume":"107","author":"Dewangan","year":"2020","journal-title":"Eng. Fail. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1061\/(ASCE)ST.1943-541X.0002812","article-title":"Modal Strain Energy-Based Model Updating Method for Damage Identification on Beam-Like Structures","volume":"146","author":"Yang","year":"2020","journal-title":"J. Struct. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"011001","DOI":"10.1115\/1.2775509","article-title":"Generalized eigenvalue decomposition in time domain modal parameter identification","volume":"130","author":"Zhou","year":"2008","journal-title":"J. Vib. Acoust."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2759","DOI":"10.21595\/jve.2017.17588","article-title":"Modal parameter identification of a three-storey structure using frequency domain techniques FDD and EFDD and time domain technique SSI: Experimental studies and simulations","volume":"19","author":"Mohammadi","year":"2017","journal-title":"J. Vibroeng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Song, Y., Liang, L., Du, Y.L., and Sun, B.C. (2020). Railway Polygonized Wheel Detection Based on Numerical Time-Frequency Analysis of Axle-Box Acceleration. Appl. Sci., 10.","DOI":"10.3390\/app10051613"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1016\/j.measurement.2019.07.027","article-title":"Application of two-dimensional wavelet transform to detect damage in steel plate structures","volume":"146","author":"Abdulkareem","year":"2019","journal-title":"Measurement"},{"key":"ref_26","first-page":"3","article-title":"Operational modal identification of structures based on improved empirical wavelet transform","volume":"26","author":"Xin","year":"2019","journal-title":"Struct. Control Health"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/j.compstruct.2018.11.042","article-title":"Crack identification of functionally graded beams using continuous wavelet transform","volume":"210","author":"Zhu","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_28","first-page":"1","article-title":"Structural Damage Identification Based on the Wavelet Transform and Improved Particle Swarm Optimization Algorithm","volume":"2020","author":"Guo","year":"2020","journal-title":"Adv. Civ. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1177\/1550147719889590","article-title":"Experimental study of grout defect identification in precast column based on wavelet packet analysis","volume":"15","author":"Zhang","year":"2019","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/j.conbuildmat.2019.03.175","article-title":"Experimental study on micro-damage identification in reinforced concrete beam with wavelet packet and DIC method","volume":"210","author":"Zhao","year":"2019","journal-title":"Constr. Build. Mater."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2859","DOI":"10.1007\/s12205-016-0523-4","article-title":"A synthesis of peak picking method and wavelet packet transform for structural modal identification","volume":"20","author":"Naderpour","year":"2016","journal-title":"KSCE J. Civ. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1111\/str.12154","article-title":"Identification of Added Mass in the Composite Plate Structure Based on Wavelet Packet Transform","volume":"52","author":"Rajendran","year":"2016","journal-title":"Strain"},{"key":"ref_33","first-page":"407","article-title":"Damage identification based on wavelet packet analysis method","volume":"52","author":"Chen","year":"2016","journal-title":"Int. J. Appl. Electromagn."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1850148","DOI":"10.1142\/S0219455418501481","article-title":"Damage Identification Using Wavelet Packet Transform and Neural Network Ensembles","volume":"18","author":"Zhang","year":"2018","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_35","first-page":"1","article-title":"Damage Identification in Structures Based on Energy Curvature Difference of Wavelet Packet Transform","volume":"2018","author":"Wang","year":"2018","journal-title":"Shock Vib."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"459","DOI":"10.12989\/sem.2013.46.4.459","article-title":"Statistical damage classification method based on wavelet packet analysis","volume":"46","author":"Law","year":"2013","journal-title":"Struct. Eng. Mech."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1061\/(ASCE)0733-9445(2008)134:5(832)","article-title":"Damage identification of shear connectors with wavelet packet energy: Laboratory test study","volume":"134","author":"Ren","year":"2008","journal-title":"J. Struct. Eng. ASCE"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2840","DOI":"10.1016\/j.engstruct.2008.03.013","article-title":"Structural damage identification by using wavelet entropy","volume":"30","author":"Ren","year":"2008","journal-title":"Eng. Struct."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"105002","DOI":"10.1088\/1361-665X\/aadaaa","article-title":"Wavelet entropy based structural damage identification under seismic excitation","volume":"27","author":"Diao","year":"2018","journal-title":"Smart Mater. Struct."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1177\/1475921714522845","article-title":"Reference-free damage detection for truss bridge structures by continuous relative wavelet entropy method","volume":"13","author":"Lee","year":"2014","journal-title":"Struct. Health Monit."},{"key":"ref_41","first-page":"1","article-title":"Damage Identification in Timber Structures Based on Wavelet Singular Spectrum Entropy","volume":"2018","author":"Li","year":"2018","journal-title":"Math. Probl. Eng."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Li, D.Y., Cao, M.S., Deng, T.F., and Zhang, S.X. (2019). Wavelet Packet Singular Entropy-Based Method for Damage Identification in Curved Continuous Girder Bridges under Seismic Excitations. Sensors, 19.","DOI":"10.3390\/s19194272"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Cao, M.S., Ding, Y.J., Ren, W.X., Wang, Q., Ragulskis, M., and Ding, Z.C. (2017). HierarchicalWavelet-Aided Neural Intelligent Identification of Structural Damage in Noisy Conditions. Appl. Sci., 7.","DOI":"10.3390\/app7040391"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.neunet.2021.02.006","article-title":"Smoothing inertial neurodynamic approach for sparse signal reconstruction via L-p-norm minimization","volume":"140","author":"Zhao","year":"2021","journal-title":"Neural Netw."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1427","DOI":"10.1016\/j.ymssp.2007.11.025","article-title":"Structural damage identification based on integration of information fusion and shannon entropy","volume":"22","author":"Li","year":"2008","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.ymssp.2019.06.027","article-title":"A novel method for single and multiple damage detection in beams using relative natural frequency changes","volume":"132","author":"Sha","year":"2019","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_47","first-page":"715","article-title":"Wavelet packet energy based damage detection index for bridge","volume":"35","author":"Zhu","year":"2015","journal-title":"J. Vib. Meas. Diagn."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"055014","DOI":"10.1088\/0964-1726\/17\/5\/055014","article-title":"Integrated wavelet transform and its application to vibration mode shapes for the damage detection of beam-type structures","volume":"17","author":"Cao","year":"2008","journal-title":"Smart Mater. Struct."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.measurement.2019.107447","article-title":"Selection of optimal wavelet-based damage-sensitive feature for seismic damage diagnosis","volume":"154","author":"Yazdanpanah","year":"2020","journal-title":"Measurement"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"106729","DOI":"10.1016\/j.ymssp.2020.106729","article-title":"Identification of instantaneous tension of bridge cables from dynamic responses: STRICT algorithm and applications","volume":"142","author":"Zhang","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"106889","DOI":"10.1016\/j.ymssp.2020.106889","article-title":"Identification of multiple cracks in noisy conditions using scale-correlation-based multiscale product of SWPT with laser vibration measurement","volume":"145","author":"Li","year":"2020","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.ymssp.2018.06.053","article-title":"Transmissibility-based system identification for structural health Monitoring: Fundamentals, approaches, and applications","volume":"117","author":"Yan","year":"2019","journal-title":"Mech. Syst. Signal Process."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/1\/239\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:55:42Z","timestamp":1760169342000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/1\/239"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,29]]},"references-count":52,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["s22010239"],"URL":"https:\/\/doi.org\/10.3390\/s22010239","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,29]]}}}