{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T18:17:49Z","timestamp":1774635469327,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T00:00:00Z","timestamp":1710460800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Jiangsu Province, China","award":["BK20200492"],"award-info":[{"award-number":["BK20200492"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Identifying the parameters of multispan rigid frames is challenging because of their complex structures and large computational workloads. This paper presents a stiffness separation method for the static response parameter identification of multispan rigid frames. The stiffness separation method segments the global stiffness matrix of the overall structure into the stiffness matrices of its substructures, which are to be computed, thereby reducing the computational workload and improving the efficiency of parameter identification. Loads can be applied individually to each separate substructure, thereby guaranteeing obvious local static responses. The veracity and efficacy of the proposed methodology are substantiated by applying it to three- and eight-span continuous rigid frame structures. The findings indicate that the proposed approach significantly enhances the efficiency of parameter identification for multispan rigid frames.<\/jats:p>","DOI":"10.3390\/s24061884","type":"journal-article","created":{"date-parts":[[2024,3,15]],"date-time":"2024-03-15T04:47:05Z","timestamp":1710478025000},"page":"1884","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Parameter Identification of Multispan Rigid Frames Using a Stiffness Separation Method"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7244-9235","authenticated-orcid":false,"given":"Feng","family":"Xiao","sequence":"first","affiliation":[{"name":"Department of Civil Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Yu","family":"Yan","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Xiangwei","family":"Meng","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Yuxue","family":"Mao","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Nanjing University of Science and Technology, Nanjing 210094, China"}]},{"given":"Gang S.","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Engineering and Computer Sciences, Marshall University, Huntington, WV 25755, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.jcsr.2011.10.009","article-title":"Dynamic stability of multi-span frames subjected to periodic loading","volume":"70","author":"Sabuncu","year":"2012","journal-title":"J. Constr. Steel Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"127571","DOI":"10.1016\/j.jclepro.2021.127571","article-title":"Curing Parameters\u2019 Influences of Early-age Temperature Field in Concrete Continuous Rigid Frame Bridge","volume":"313","author":"Zeng","year":"2021","journal-title":"J. Clean. Prod."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"102844","DOI":"10.1016\/j.marstruc.2020.102844","article-title":"Analysis of collapse resistance of offshore rigid frame\u2014Continuous girder bridge based on time-varying fragility","volume":"75","author":"Liang","year":"2021","journal-title":"Mar. Struct."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"114838","DOI":"10.1016\/j.engstruct.2022.114838","article-title":"Innovative design of long-span steel\u2013concrete composite bridge using multi-material topology optimization","volume":"269","author":"Li","year":"2022","journal-title":"Eng. Struct."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0958-9465(99)00047-5","article-title":"Construction of Benten Viaduct, rigid-frame bridge with seismic isolators at the foot of piers","volume":"22","author":"Yoshikawa","year":"2000","journal-title":"Cem. Concr. Compos."},{"key":"ref_6","first-page":"343","article-title":"Unified calculation model for the longitudinal fundamental frequency of continuous rigid frame bridge","volume":"3","author":"Zhou","year":"2021","journal-title":"Struct. Eng. Mech."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Liu, W., Lai, H., Dai, G., Rao, S., Wang, D., and Wu, B. (2021). Numerical study on track\u2013bridge interaction of integral railway rigid-frame bridge. Appl. Sci., 11.","DOI":"10.3390\/app11030922"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"442","DOI":"10.1016\/j.advengsoft.2009.03.005","article-title":"Cost optimization of industrial steel building structures","volume":"41","author":"Kravanja","year":"2010","journal-title":"Adv. Eng. Softw."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/j.soildyn.2018.07.047","article-title":"Analysis of seismic non-structural damage in single-storey industrial steel buildings","volume":"114","author":"Scozzese","year":"2018","journal-title":"Soil Dyn. Earthq. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.jcsr.2012.11.008","article-title":"Optimal design of single-story steel building structures based on parametric MINLP optimization","volume":"81","author":"Kravanja","year":"2013","journal-title":"J. Constr. Steel Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1833","DOI":"10.1016\/j.istruc.2023.03.152","article-title":"Damage identification in steel frames using dual-criteria vibration-based damage detection method and artificial neural network","volume":"51","author":"Nick","year":"2023","journal-title":"Structures"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"e2288","DOI":"10.1002\/stc.2288","article-title":"Damage detection in structural systems utilizing artificial neural networks and proper orthogonal decomposition","volume":"26","author":"Azam","year":"2019","journal-title":"Struct. Control Health Monit."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.jcsr.2017.10.023","article-title":"Health monitoring of a steel moment-resisting frame subjected to seismic loads","volume":"140","author":"Mosallam","year":"2018","journal-title":"J. Constr. Steel Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1177\/1475921718804132","article-title":"A novel deep learning-based method for damage identification of smart building structures","volume":"18","author":"Yu","year":"2018","journal-title":"Struct. Health Monit."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1177\/1687814019832216","article-title":"Bridge health monitoring and damage identification of truss bridge using strain measurements","volume":"11","author":"Xiao","year":"2019","journal-title":"Adv. Mech. Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"6631716","DOI":"10.1155\/2023\/6631716","article-title":"Parameter identification of frame structures by considering shear deformation","volume":"2023","author":"Xiao","year":"2023","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"110518","DOI":"10.1016\/j.ymssp.2023.110518","article-title":"Damage identification method for tied arch bridge suspender based on quasi-static displacement influence line","volume":"200","author":"Fan","year":"2023","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"106142","DOI":"10.1016\/j.compstruc.2019.106142","article-title":"A fast friction-model-inspired sparse regularization approach for damage identification with modal data","volume":"227","author":"Wang","year":"2020","journal-title":"Comput. Struct."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"107142","DOI":"10.1016\/j.engfailanal.2023.107142","article-title":"Modal strain energy based enhanced approaches for damage detection and severity estimation","volume":"146","author":"Nguyen","year":"2023","journal-title":"Eng. Fail. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"105661","DOI":"10.1016\/j.ijfatigue.2020.105661","article-title":"Thermoelasticity-based modal damage identification","volume":"137","author":"Capponi","year":"2020","journal-title":"Int. J. Fatigue"},{"key":"ref_22","first-page":"555","article-title":"Parameter estimation of structures from static strain measurements","volume":"122","author":"Sanayei","year":"1996","journal-title":"I: Formulation Struct. Eng."},{"key":"ref_23","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."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1550147717707929","DOI":"10.1177\/1550147717707929","article-title":"Optimal static strain sensor placement for truss bridges","volume":"13","author":"Xiao","year":"2017","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"112599","DOI":"10.1016\/j.measurement.2023.112599","article-title":"Structural damage detection of the bridge under moving loads with the quasi-static displacement influence line from one sensor","volume":"211","author":"Zhu","year":"2023","journal-title":"Measurement"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1260\/136943307783571409","article-title":"Parameter identification and damage detection using structural optimization and static response data","volume":"10","author":"Terlaje","year":"2007","journal-title":"Adv. Struct. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1080\/17415977.2016.1169277","article-title":"Structural damage diagnosis using incomplete static responses and LS-SVM","volume":"25","author":"Kourehli","year":"2016","journal-title":"Inverse Probl. Sci. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"116161","DOI":"10.1016\/j.jsv.2021.116161","article-title":"Model updating for a large multi-span quasi-periodic viaduct based on free wave characteristics","volume":"506","author":"Zhang","year":"2021","journal-title":"J. Sound Vib."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"105016","DOI":"10.1016\/j.istruc.2023.105016","article-title":"Structural safety assessment oriented modal experiments on Renyihe Bridge using vehicle excitations","volume":"56","author":"Cheng","year":"2023","journal-title":"Structures"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"109908","DOI":"10.1016\/j.ymssp.2022.109908","article-title":"Damage identification of long-span bridges based on the correlation of probability distribution of monitored quasi-static responses","volume":"186","author":"Deng","year":"2023","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"209","DOI":"10.12989\/sss.2014.14.2.209","article-title":"Localized reliability analysis on a large-span rigid frame bridge based on monitored strains from the long-term SHM system","volume":"14","author":"Liu","year":"2014","journal-title":"Smart Struct. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1016\/j.jsv.2009.01.015","article-title":"Improved substructuring method for eigensolutions of large-scale structures","volume":"323","author":"Weng","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.istruc.2023.04.027","article-title":"Damage identification of large-scale space truss structures based on stiffness separation method","volume":"53","author":"Xiao","year":"2023","journal-title":"Structures"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/j.jsv.2005.04.006","article-title":"Damage detection by finite element model updating using modal flexibility residual","volume":"290","author":"Jaishi","year":"2006","journal-title":"J. Sound Vib."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1174","DOI":"10.2514\/3.10720","article-title":"Damage assessment of structures using static test data","volume":"29","author":"Sanayei","year":"2001","journal-title":"AIAA J."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Xiao, F., Zhu, W., Meng, X., and Chen, G.S. (2022). Parameter identification of structures with different connections using static responses. Appl. Sci., 12.","DOI":"10.3390\/app12125896"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/S0141-0296(00)00067-5","article-title":"Joint damage assessment of framed structures using a neural networks technique","volume":"23","author":"Yun","year":"2001","journal-title":"Eng. Struct."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1260\/1369-4332.17.8.1075","article-title":"Damage Identification of frame structures with joint damage under earthquake excitation","volume":"17","author":"Lei","year":"2014","journal-title":"Adv. Struct. Eng."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.1061\/(ASCE)ST.1943-541X.0000079","article-title":"Experimental study of damage detection by data-driven subspace identification and finite-element model updating","volume":"135","author":"Weng","year":"2009","journal-title":"J. Struct. Eng."},{"key":"ref_40","first-page":"407","article-title":"Modal parameter based inverse approach for structural joint damage assessment using unified particle swarm optimization","volume":"242","author":"Nanda","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1007\/s13349-020-00390-8","article-title":"Combined joint and member damage identification of skeletal structures by an improved biology migration algorithm","volume":"10","author":"Aval","year":"2020","journal-title":"J. Civ. Struct. Health Monit."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1057\/jors.2014.103","article-title":"A better measure of relative prediction accuracy for model selection and model estimation","volume":"66","author":"Tofallis","year":"2015","journal-title":"J. Oper. Res. Soc."},{"key":"ref_43","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."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/6\/1884\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:14:07Z","timestamp":1760105647000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/6\/1884"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,15]]},"references-count":43,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["s24061884"],"URL":"https:\/\/doi.org\/10.3390\/s24061884","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,15]]}}}