{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T00:09:07Z","timestamp":1782173347435,"version":"3.54.5"},"reference-count":36,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2021,11,4]],"date-time":"2021-11-04T00:00:00Z","timestamp":1635984000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Natural Science Foundations of China","award":["U1934207, 52078487, and 51778630"],"award-info":[{"award-number":["U1934207, 52078487, and 51778630"]}]},{"name":"the Hunan Innovative Provincial Construction Project","award":["2019RS3009"],"award-info":[{"award-number":["2019RS3009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Materials"],"abstract":"<jats:p>This paper examines the effect of structural deformation on the unit slab-type ballastless track structure of high-speed railway. The principle of stationary potential energy was used to map the relation between girder vertical deformation and rail deformation considering the effect of subgrade boundary conditions and the nonlinear contact of interlayer. The theoretical model was verified by comparing with the finite element analysis and experimental results. The theoretical model was used to analyze the effects of several key parameters on the rail deformation, such as vertical deformation amplitude, elastic modulus of the mortar layer, and vertical stiffness of the fasteners. The results show that the track slabs suffered significant disengagement, which makes the deformation of the track structure at the position of the beam joint tend to be gentle when nonlinear contact between the mortar layer and the track slabs was considered. The track slabs disengagement mainly occurs near the beam joints (the side of the deformed beam). As the deflection amplitude of the girder increases, the track deformation, the fastener forces and the disengagement length of the track slabs are obviously nonlinear. When the vertical stiffness of the fastener and\/or the elastic modulus of the mortar layer increase, the fastener force and the track plate disengagement length increase monotonically and nonlinearly, which will adversely affect the life and safety of the track structure.<\/jats:p>","DOI":"10.3390\/ma14216653","type":"journal-article","created":{"date-parts":[[2021,11,4]],"date-time":"2021-11-04T22:25:54Z","timestamp":1636064754000},"page":"6653","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Mapping Relation between Rail and Bridge Deformation Considering Nonlinear Contact of Interlayer"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4405-0689","authenticated-orcid":false,"given":"Leixin","family":"Nie","sequence":"first","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lizhong","family":"Jiang","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"},{"name":"National Engineering Laboratory for High-Speed Railway Construction, Central South University, Changsha 410075, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3338-4971","authenticated-orcid":false,"given":"Wangbao","family":"Zhou","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, Central South University, Changsha 410075, China"},{"name":"National Engineering Laboratory for High-Speed Railway Construction, Central South University, Changsha 410075, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yulin","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Civil Engineering and Architecture, East China Jiaotong University, Nanchang 330013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2050120","DOI":"10.1142\/S0219455420501205","article-title":"Design and development of smart semi active suspension for nonlinear rail vehicle vibration reductio","volume":"20","author":"Sharma","year":"2020","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1177\/1369433219896166","article-title":"Seismic damage features of high-speed railway simply supported bridge-track system under near-fault earthquake","volume":"23","author":"Guo","year":"2020","journal-title":"Adv. Struct. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.engstruct.2014.04.018","article-title":"Probabilistic safety assessment of a short span high-speed railway bridge","volume":"71","author":"Rocha","year":"2014","journal-title":"Eng. Struct."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1177\/1369433219875304","article-title":"Efficient evaluation of bridge deformation for running safety of railway vehicles using simplified models","volume":"23","author":"Jin","year":"2020","journal-title":"Adv. Struct. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.conbuildmat.2015.08.024","article-title":"A study of the dynamic mechanical properties of CRTS I type CA mortar","volume":"112","author":"Zeng","year":"2016","journal-title":"Constr. Build. Mater."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Shan, Y., Zheng, S., Zhang, X., Luo, W., Mao, J., and Kong, D. (2018). Fatigue performance of the CA mortar used in CRTS i ballastless slab track under simulated servicing condition. Materials, 11.","DOI":"10.3390\/ma11112259"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1950111","DOI":"10.1142\/S0219455419501116","article-title":"Effect of Long-Term bridge deformations on safe operation of High-Speed railway and vibration of vehicle-bridge coupled system","volume":"19","author":"Gou","year":"2019","journal-title":"Int. J. Struct. Stab. Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1260\/1369-4332.17.5.709","article-title":"Dynamic analysis of Train-Rail-Bridge interaction considering concrete creep of a Multi-Span simply supported bridge","volume":"17","author":"Yang","year":"2014","journal-title":"Adv. Struct. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.engstruct.2017.08.057","article-title":"Seismic damage evaluation of high-speed railway bridge components under different intensities of earthquake excitations","volume":"152","author":"Kang","year":"2017","journal-title":"Eng. Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2969","DOI":"10.1016\/j.engstruct.2010.05.016","article-title":"Post-yield stiffnesses and residual deformations of RC bridge columns reinforced with ordinary rebars and steel fiber composite bars","volume":"32","author":"Fahmy","year":"2010","journal-title":"Eng. Struct."},{"key":"ref_11","first-page":"74","article-title":"Mechanical characterization of railway structures based on vertical stiffness analysis and railway substructure stress state","volume":"227","author":"Real","year":"2012","journal-title":"Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1080\/15732479.2019.1594314","article-title":"Dynamic behavior of hybrid framed arch railway bridge under moving trains","volume":"15","author":"Gou","year":"2019","journal-title":"Struct. Infrastruct. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1061\/(ASCE)BE.1943-5592.0000506","article-title":"Safety of maglev trains moving on bridges subject to foundation settlements and earthquakes","volume":"19","author":"Ju","year":"2014","journal-title":"J. Bridge Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.engstruct.2014.04.037","article-title":"Influence of the vehicle model on the prediction of the maximum bending response of simply-supported bridges under high-speed railway traffic","volume":"72","author":"Museros","year":"2014","journal-title":"Eng. Struct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/s00419-012-0653-1","article-title":"3D analysis of high-speed trains moving on bridges with foundation settlements","volume":"83","author":"Ju","year":"2013","journal-title":"Arch. Appl. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2115","DOI":"10.1016\/j.engstruct.2009.03.019","article-title":"Response of a train moving on multi-span railway bridges undergoing ground settlement","volume":"31","author":"Yau","year":"2009","journal-title":"Eng. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"680","DOI":"10.1061\/(ASCE)GT.1943-5606.0000032","article-title":"Dynamic stress analysis of a ballasted railway track bed during train passage","volume":"135","author":"Yang","year":"2009","journal-title":"J. Geotech. Geoenviron. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.trgeo.2018.10.007","article-title":"Geometry mapping and additional stresses of ballastless track structure caused by subgrade differential settlement under self-weight loads in high-speed railways","volume":"18","author":"Jiang","year":"2019","journal-title":"Transp. Geotechnol."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gou, H., Zhou, W., Bao, Y., Li, X., and Pu, Q. (2018). Experimental study on dynamic effects of a long-span railway continuous beam bridge. Appl. Sci., 8.","DOI":"10.3390\/app8050669"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"936","DOI":"10.1177\/0954409717702019","article-title":"Dynamic performance of continuous railway bridges: Numerical analyses and field tests","volume":"232","author":"Gou","year":"2018","journal-title":"Proc. Inst. Mech. Eng. Part F-J. Rail Rapid Transit"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1177\/0954409718766929","article-title":"Experimental and numerical investigations of the dynamic responses of an asymmetrical arch railway bridge","volume":"232","author":"Gou","year":"2018","journal-title":"Proc. Inst. Mech. Eng. Part F-J. Rail Rapid Transit"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1177\/0954409716675001","article-title":"Analysis of structural stresses of tracks and vehicle dynamic responses in train-track-bridge system with pier settlement","volume":"232","author":"Chen","year":"2018","journal-title":"Proc. Inst. Mech. Eng. Part F-J. Rail Rapid Transit"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1016\/S0022-460X(03)00782-X","article-title":"A theoretical model for ground vibration from trains generated by vertical track irregularities","volume":"272","author":"Sheng","year":"2004","journal-title":"J. Sound Vib."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1007\/s11431-014-5692-0","article-title":"Safety threshold of high-speed railway pier settlement based on train-track-bridge dynamic interaction","volume":"58","author":"Chen","year":"2015","journal-title":"Sci. China-Technol. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1360\/N092014-00105","article-title":"Mapping relationship between pier settlement and rail deformation of high-speed railways\u2014Part (I): The unit slab track system","volume":"44","author":"Chen","year":"2014","journal-title":"Sci. Sin. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1080\/15732479.2017.1359189","article-title":"Study on the safe value of multi-pier settlement for simply supported girder bridges in high-speed railways","volume":"14","author":"Chen","year":"2018","journal-title":"Struct. Infrastruct. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1360\/N092014-00106","article-title":"Mapping relationship between pier settlement and rail deformation of high-speed railways\u2014Part (II): The longitudinal connected ballastless track system","volume":"44","author":"Chen","year":"2014","journal-title":"Sci. Sin. Technol."},{"key":"ref_28","first-page":"219","article-title":"Effect of bridge lateral deformation on track geometry of high-speed railway","volume":"29","author":"Gou","year":"2018","journal-title":"Steel Compos. Struct."},{"key":"ref_29","first-page":"467","article-title":"Mapping vertical bridge deformations to track geometry for high-speed railway","volume":"32","author":"Gou","year":"2019","journal-title":"Steel Compos. Struct."},{"key":"ref_30","first-page":"209","article-title":"An analytical solution to the mapping relationship between bridge structures vertical deformation and rail deformation of high-speed railway","volume":"32","author":"Feng","year":"2019","journal-title":"Steel Compos. Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1016\/j.istruc.2020.07.023","article-title":"Mapping relation between pier settlement and rail deformation of unit slab track system","volume":"27","author":"Zhou","year":"2020","journal-title":"Structures"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1177\/0954409719880668","article-title":"Mapping the relationship between the structural deformation of a simply supported beam bridge and rail deformation in high-speed railways","volume":"234","author":"Jiang","year":"2020","journal-title":"Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit"},{"key":"ref_33","first-page":"15","article-title":"A mechanical model of vehicle-slab track coupled system with differential subgrade settlement","volume":"66","author":"Guo","year":"2018","journal-title":"Struct. Eng. Mech."},{"key":"ref_34","first-page":"286","article-title":"Effect of unsupported sleepers on rail track dynamic behaviour","volume":"171","author":"Sadeghi","year":"2018","journal-title":"Proc. Inst. Civ. Eng.-Transp."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Sysyn, M., Przybylowicz, M., Nabochenko, O., and Liu, J. (2021). Mechanism of Sleeper-Ballast dynamic impact and residual settlements accumulation in zones with unsupported sleepers. Sustainability, 13.","DOI":"10.20944\/preprints202107.0066.v1"},{"key":"ref_36","unstructured":"Wei, Y. (2012). Research on Beam End Displacement Relevant Problems for High-Speed Railway Ballastless Track Bridge, China Academy of Railway Sciences. (In Chinese)."}],"container-title":["Materials"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1996-1944\/14\/21\/6653\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:25:59Z","timestamp":1760167559000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1996-1944\/14\/21\/6653"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,4]]},"references-count":36,"journal-issue":{"issue":"21","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["ma14216653"],"URL":"https:\/\/doi.org\/10.3390\/ma14216653","relation":{},"ISSN":["1996-1944"],"issn-type":[{"value":"1996-1944","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,4]]}}}