{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:14:36Z","timestamp":1760145276332,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,28]],"date-time":"2024-06-28T00:00:00Z","timestamp":1719532800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, the vibration analysis of a layered, cylinder-shaped shell is undertaken. The structure of the shell layers is composed of functionally graded and isotropic materials. The vibrations of four-layered cylindrical shells with a ring support along the axial direction are investigated in this research. The two internal layers are composed of isotropic materials, and the external two layers are composed of functionally graded materials. The outer functionally graded material layers considered are stainless steel, zirconia, and nickel. The inner two isotropic layers considered are aluminum and stainless steel. The shell frequency equation is acquired by employing the Rayleigh\u2013Ritz method under the shell theory of Sanders. The trigonometric volume fraction law is used to sort the functionally graded material composition of the FGM layers. The natural frequencies are attained under two boundary conditions, namely simply supported\u2013simply supported and clamped\u2013clamped.<\/jats:p>","DOI":"10.3390\/sym16070812","type":"journal-article","created":{"date-parts":[[2024,7,3]],"date-time":"2024-07-03T04:23:43Z","timestamp":1719980623000},"page":"812","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Structural Study of Four-Layered Cylindrical Shell Comprising Ring Support"],"prefix":"10.3390","volume":"16","author":[{"given":"Madiha","family":"Ghamkhar","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5559-5951","authenticated-orcid":false,"given":"Ahmad","family":"N. Al-Kenani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Naveed","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Agriculture, Faisalabad 38000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"257","DOI":"10.6089\/jscm.13.257","article-title":"Functionally gradient materials. In pursuit of super heat-resisting materials for spacecraft","volume":"13","author":"Niino","year":"1987","journal-title":"J. Jpn. Soc. Compos. Mater."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1006\/jsvi.1995.0599","article-title":"Effects of boundary conditions on frequencies of a multi-layered cylindrical shell","volume":"188","author":"Lam","year":"1995","journal-title":"J. Sound Vib."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1006\/jsvi.1995.0503","article-title":"Vibrations of cylindrical shells with intermediate supports","volume":"187","author":"Swaddiwudhipong","year":"1995","journal-title":"J. Sound Vib."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/S0020-7403(96)00035-5","article-title":"Vibration of cylindrical shells with ring support","volume":"39","author":"Loy","year":"1997","journal-title":"Int. J. Mech. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/S0020-7403(98)00054-X","article-title":"Vibration of functionally graded cylindrical shells","volume":"41","author":"Loy","year":"1999","journal-title":"Int. J. Mech. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/S0003-682X(99)00063-8","article-title":"Vibration characteristics of functionally graded cylindrical shells under various boundary conditions","volume":"61","author":"Pradhan","year":"2000","journal-title":"Appl. Acoust."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1907","DOI":"10.1016\/S0020-7403(02)00071-1","article-title":"Exact solutions for vibration of cylindrical shells with intermediate ring supports","volume":"44","author":"Xiang","year":"2002","journal-title":"Int. J. Mech. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.ijmecsci.2003.12.005","article-title":"Three-dimensional vibration analysis of fluid-filled orthotropic FGM cylindrical shells","volume":"46","author":"Chen","year":"2004","journal-title":"Int. J. Mech. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3705","DOI":"10.1016\/j.ijsolstr.2005.05.058","article-title":"Vibration of open circular cylindrical shells with intermediate ring supports","volume":"43","author":"Zhang","year":"2006","journal-title":"Int. J. Solids Struct."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.1016\/j.ijmecsci.2006.05.005","article-title":"Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions","volume":"48","author":"Zhang","year":"2006","journal-title":"Int. J. Mech. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.jsv.2007.05.019","article-title":"Free vibration of FGM cylindrical shells with holes under various boundary conditions","volume":"306","year":"2007","journal-title":"J. Sound Vib."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s00707-009-0141-z","article-title":"Vibration characteristics of FGM circular cylindrical shells using wave propagation approach","volume":"208","author":"Iqbal","year":"2009","journal-title":"Acta Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1007\/s10483-009-1105-x","article-title":"Vibration characteristics of FGM circular cylindrical shells filled with fluid using wave propagation approach","volume":"30","author":"Iqbal","year":"2009","journal-title":"Appl. Math. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/j.mechrescom.2010.07.006","article-title":"Free vibration of three-layer circular cylindrical shells with functionally graded middle layer","volume":"37","author":"Li","year":"2010","journal-title":"Mech. Res. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"228","DOI":"10.4236\/eng.2010.24033","article-title":"The stability of cylindrical shells containing an FGM layer subjected to axial load on the Pasternak foundation","volume":"2","author":"Sofiyev","year":"2010","journal-title":"Engineering"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s12206-010-0122-0","article-title":"Vibration of bilayered cylindrical shells with layers of different materials","volume":"24","author":"Arshad","year":"2010","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"315","DOI":"10.4236\/wjm.2014.411032","article-title":"Vibration of three-layered FGM cylindrical shells with a middle layer of isotropic material for various boundary conditions","volume":"4","author":"Naeem","year":"2014","journal-title":"World J. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"9","DOI":"10.7603\/s40632-016-0009-z","article-title":"Influence of ring support on free vibration of sandwich functionally graded cylindrical shells with a middle layer of isotropic material","volume":"4","author":"Arshad","year":"2016","journal-title":"J. Eng. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1566","DOI":"10.1038\/s41598-018-38122-0","article-title":"Vibration frequency analysis of three-layered cylinder-shaped shell with the effect of FGM central layer thickness","volume":"9","author":"Ghamkhar","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1515\/phys-2019-0060","article-title":"Vibration analysis of a three-layered FGM cylindrical shell including the effect of ring support","volume":"17","author":"Ghamkhar","year":"2019","journal-title":"Open Phys."},{"doi-asserted-by":"crossref","unstructured":"Liu, Z., Tornabene, F., Dimitri, R., and Babaei, M. (2024). Numerical Study of the Buckling Response of Stiffened FG Graphene-Reinforced Multilayer Composite Cylindrical Panels. Processes, 12.","key":"ref_21","DOI":"10.3390\/pr12030430"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/7\/812\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:07:05Z","timestamp":1760108825000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/7\/812"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,28]]},"references-count":21,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["sym16070812"],"URL":"https:\/\/doi.org\/10.3390\/sym16070812","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,6,28]]}}}