Semi-Analytical Analysis of Thin Plate Deflections Using the Clebsch–Ritz Method

Semi-Analytical Analysis of Thin Plate Deflections Using the Clebsch–Ritz Method | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Semi-Analytical Analysis of Thin Plate Deflections Using the Clebsch–Ritz Method Hemza Gherdaoui, Guenfoud Salah, Hadjira. Bouchelaghem, Benmohamed Hanene, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8029987/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Plates of various geometries and materials are widely employed in structural and mechanical engineering, where accurate prediction of their deformation behavior is crucial for design and safety. This study presents a semi-analytical approach for determining the deflection of thin rectangular plates resting on an elastic foundation with free boundary conditions. The proposed method integrates Clebsch’s analytical solution of the plate deformation equation with the Ritz energy method to compute the unknown coefficients governing the displacement field. By combining the strain energy of deformation with the external work of the applied load, the governing variational equations are established and solved to determine the deformation parameters. The developed formulation allows direct evaluation of the plate deflection at any point for arbitrary loading conditions. Numerical implementation using *Mathematica* validates the accuracy of the model and confirms its efficiency in representing the physical behavior of plates on elastic media. The proposed Clebsch–Ritz approach offers a reliable and adaptable framework for analyzing static deformation problems in structural and mechanical applications. Plate’s Deflections Ritz’s Method Clebsch’s Solution Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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Method\u003c/strong\u003e\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"8 May 1945 University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Plate’s Deflections, Ritz’s Method, Clebsch’s Solution","lastPublishedDoi":"10.21203/rs.3.rs-8029987/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8029987/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePlates of various geometries and materials are widely employed in structural and mechanical engineering, where accurate prediction of their deformation behavior is crucial for design and safety. This study presents a semi-analytical approach for determining the deflection of thin rectangular plates resting on an elastic foundation with free boundary conditions. The proposed method integrates Clebsch\u0026rsquo;s analytical solution of the plate deformation equation with the Ritz energy method to compute the unknown coefficients governing the displacement field. By combining the strain energy of deformation with the external work of the applied load, the governing variational equations are established and solved to determine the deformation parameters. The developed formulation allows direct evaluation of the plate deflection at any point for arbitrary loading conditions. Numerical implementation using *Mathematica* validates the accuracy of the model and confirms its efficiency in representing the physical behavior of plates on elastic media. The proposed Clebsch\u0026ndash;Ritz approach offers a reliable and adaptable framework for analyzing static deformation problems in structural and mechanical applications.\u003c/p\u003e","manuscriptTitle":"Semi-Analytical Analysis of Thin Plate Deflections Using the Clebsch–Ritz Method","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-04 05:34:52","doi":"10.21203/rs.3.rs-8029987/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"5c49644f-667c-4d5b-9432-c4d4700d1bbb","owner":[],"postedDate":"December 4th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-12-04T05:34:52+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-04 05:34:52","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8029987","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8029987","identity":"rs-8029987","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}