A Finite Difference Approach to the Buckling Analysis of Fully Embedded Piles under Various End Restraints

A Finite Difference Approach to the Buckling Analysis of Fully Embedded Piles under Various End Restraints | 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 A Finite Difference Approach to the Buckling Analysis of Fully Embedded Piles under Various End Restraints Vlora Shatri, Burbuqe Shatri, Bajram Shefkiu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8991385/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 This paper presents an analytical investigation into the buckling behaviour of slender piles subjected to axial loading, incorporating the effects of nonlinear pile–soil interaction modeled through the Winkler foundation approach. The soil reaction is characterized by a depth-dependent modulus of subgrade reaction to represent the variation of soil stiffness with depth. The governing differential equation is discretized using the Finite Difference Method (FDM) and transformed into an eigenvalue problem, which is solved numerically in MATLAB to determine the critical buckling loads and associated mode shapes. A detailed parametric analysis is conducted to evaluate the influence of pile length, soil stiffness parameters, and boundary conditions on overall pile stability. The results demonstrate that the stiffness variation of the supporting soil has a dominant effect on the buckling resistance compared to the initial stiffness value. Furthermore, the study reveals that piles with higher restraint conditions exhibit greater buckling capacity, while mode shape transitions observed with increasing pile length highlight the nonlinear nature of the system. The findings emphasize the necessity of incorporating realistic soil stiffness variation and boundary conditions in the design and assessment of pile foundations to ensure structural safety and reliability. Slender piles Buckling analysis Finite Difference Method Pile soil interaction Boundary Full Text Additional Declarations No competing interests reported. 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. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8991385","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":601055157,"identity":"862428e9-1c58-4f94-a661-16154114424c","order_by":0,"name":"Vlora Shatri","email":"","orcid":"","institution":"University of Prishtina","correspondingAuthor":false,"prefix":"","firstName":"Vlora","middleName":"","lastName":"Shatri","suffix":""},{"id":601055158,"identity":"382c94e9-25d7-46f8-a31c-453f327c07f8","order_by":1,"name":"Burbuqe Shatri","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVRIiWNgGAWjYBACAwYGNghLAsy1YWBghvAZG4jQAlRkkEayFobDcBmcWswlcp895imzYZCf3fz8cUXB+cTt7AxsjysYbGQ3HMCuxXJGurkxz7k0BoM7xwwbzxjcTtzZzMBueIYhzRiXFoMbaWzSvG2HGQwkEgwbG4BaNhzm/ybZwHA4kYCW/wzyM9I/ArWcA2phYANq+U9IC1D2Rg7IlgMwLQdwarHsecZuOOdcMo/BjZzCmQ0GycZALeyGIMZMHFrM2dPYHrwps5MDOmzDx4Y/drIbzh9ge9hQYSfbh0MLBLAx8KByQfGFH7Dh5Y6CUTAKRsFIBwCrEFq9EqhcQAAAAABJRU5ErkJggg==","orcid":"","institution":"University of Prishtina","correspondingAuthor":true,"prefix":"","firstName":"Burbuqe","middleName":"","lastName":"Shatri","suffix":""},{"id":601055159,"identity":"b52b6695-3df2-4291-8f24-8b62ca061a11","order_by":2,"name":"Bajram Shefkiu","email":"","orcid":"","institution":"University of Prishtina","correspondingAuthor":false,"prefix":"","firstName":"Bajram","middleName":"","lastName":"Shefkiu","suffix":""}],"badges":[],"createdAt":"2026-02-27 20:13:23","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8991385/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8991385/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104237710,"identity":"afa6cd36-4637-4849-819c-24a47d3b9985","added_by":"auto","created_at":"2026-03-09 13:42:41","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":728672,"visible":true,"origin":"","legend":"","description":"","filename":"Article22.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8991385/v1_covered_806ee43b-e020-430b-992a-174b890b6b06.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Finite Difference Approach to the Buckling Analysis of Fully Embedded Piles under Various End Restraints","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":true,"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":"Slender piles, Buckling analysis, Finite Difference Method, Pile, soil interaction, Boundary","lastPublishedDoi":"10.21203/rs.3.rs-8991385/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8991385/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper presents an analytical investigation into the buckling behaviour of slender piles subjected to axial loading, incorporating the effects of nonlinear pile\u0026ndash;soil interaction modeled through the Winkler foundation approach. The soil reaction is characterized by a depth-dependent modulus of subgrade reaction to represent the variation of soil stiffness with depth. The governing differential equation is discretized using the Finite Difference Method (FDM) and transformed into an eigenvalue problem, which is solved numerically in MATLAB to determine the critical buckling loads and associated mode shapes. A detailed parametric analysis is conducted to evaluate the influence of pile length, soil stiffness parameters, and boundary conditions on overall pile stability. The results demonstrate that the stiffness variation of the supporting soil has a dominant effect on the buckling resistance compared to the initial stiffness value. Furthermore, the study reveals that piles with higher restraint conditions exhibit greater buckling capacity, while mode shape transitions observed with increasing pile length highlight the nonlinear nature of the system. 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