{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T18:07:16Z","timestamp":1775153236160,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T00:00:00Z","timestamp":1763337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>An analytical method is proposed for determining the stress-strain state in an elastic layer with a cylindrical cavity supported by linear continuous supports perpendicular to the cavity. The need for such a development is due to the fact that in aerospace and mechanical engineering, structural elements are often affected by loads and supports described by infinite functions. This complicates the calculation for spatial bodies with complex geometry and stress concentrators. The methodology is based on the generalized Fourier method within the spatial problem of elasticity theory. The model is considered as a layer with specified stresses at the outer boundaries, where the reactions of the supports are represented as applied loads. A combined approach is used to describe the geometry using a Cartesian coordinate system for the layer and a cylindrical coordinate system for the cavity. The key idea is to decompose the original problem into two simpler ones using the principle of superposition. Auxiliary problem: the stresses in a solid layer (without a cavity) are calculated to determine the stress fields at its nominal location. Main problem: a layer with a cavity is considered, on the surface of which the stresses calculated in the first step are acting but taken with the opposite sign. The complete solution is the sum of the solutions of these two problems. Each of them is reduced to an infinite system of linear algebraic equations, which is solved by the method of reduction. This approach makes it possible to calculate the stress-strain state at any point of the body with high accuracy. Numerical analysis confirmed the correctness of satisfying the boundary conditions and showed the dependence of stresses on the nature of the distributed loads. The cylindrical cavity acts as a stress concentrator, which leads to a local increase in stresses \u03c3x and \u03c3z at the upper and lower boundaries of the layer to values that exceed both the applied load by and the calculated resistance of concrete of class C25\/30.<\/jats:p>","DOI":"10.3390\/computation13110270","type":"journal-article","created":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T15:38:09Z","timestamp":1763393889000},"page":"270","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Consideration of Spatially Infinite Loads in the Problem for a Layer with a Cylindrical Cavity and Continuous Supports"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7406-5809","authenticated-orcid":false,"given":"Nataliia","family":"Ukrayinets","sequence":"first","affiliation":[{"name":"Department of Higher Mathematics and Systems Analysis, National Aerospace University \u201cKharkiv Aviation Institute\u201d, 61070 Kharkiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7234-1558","authenticated-orcid":false,"given":"Tetyana","family":"Alyoshechkina","sequence":"additional","affiliation":[{"name":"Department of Theoretical and Construction Mechanics, O.M. Beketov National University of Urban Economy in Kharkiv, 61002 Kharkiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9491-0181","authenticated-orcid":false,"given":"Vitaly","family":"Miroshnikov","sequence":"additional","affiliation":[{"name":"Department of Aircraft Strength, National Aerospace University \u201cKharkiv Aviation Institute\u201d, 61070 Kharkiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2664-0255","authenticated-orcid":false,"given":"Oleksandr","family":"Savin","sequence":"additional","affiliation":[{"name":"Department of Aircraft Strength, National Aerospace University \u201cKharkiv Aviation Institute\u201d, 61070 Kharkiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5693-6954","authenticated-orcid":false,"given":"Basheer","family":"Younis","sequence":"additional","affiliation":[{"name":"Department of Aircraft Strength, National Aerospace University \u201cKharkiv Aviation Institute\u201d, 61070 Kharkiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4492-3862","authenticated-orcid":false,"given":"Vitalii","family":"Vynohradov","sequence":"additional","affiliation":[{"name":"Department of Theoretical and Construction Mechanics, O.M. Beketov National University of Urban Economy in Kharkiv, 61002 Kharkiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6170-5173","authenticated-orcid":false,"given":"Olena","family":"Murahovska","sequence":"additional","affiliation":[{"name":"Department of Higher Mathematics and Systems Analysis, National Aerospace University \u201cKharkiv Aviation Institute\u201d, 61070 Kharkiv, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s40799-021-00439-3","article-title":"Elastic properties of steel-cord rubber conveyor belt","volume":"45","author":"Golovanevskiy","year":"2021","journal-title":"Exp. Tech."},{"key":"ref_2","unstructured":"Kuhn, H., and Medlin, D. (2023). ASM Handbook, Volume 8: Mechanical Testing and Evaluation, ASM International."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s11029-019-09808-x","article-title":"Relationships Between the Ultimate Strengths of Polymer Composites in Static Bending, Compression, and Tension","volume":"55","author":"Kondratiev","year":"2019","journal-title":"Mech. Compos. Mater."},{"key":"ref_4","first-page":"107","article-title":"Physico-mechanical characteristics of nano-composite materials based on fluoroplastic","volume":"3","author":"Revo","year":"2021","journal-title":"Bull. Taras Shevchenko Natl. Univ. Kyiv. Phys. Math. Sci."},{"key":"ref_5","unstructured":"Zienkiewicz, O.C. (1971). The Finite Element Method in Engineering Science, McGraw-Hill."},{"key":"ref_6","unstructured":"Zienkiewicz, O.C., Taylor, R.L., and Zhu, J.Z. (2013). The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann. [7th ed.]."},{"key":"ref_7","unstructured":"Ansys (2025, August 25). Static Structural Simulation Using Ansys Discovery. Available online: https:\/\/courses.ansys.com\/index.php\/courses\/structural-simulation."},{"key":"ref_8","first-page":"67","article-title":"Mathematical modeling of the dynamics of an elastic half-medium with a cylindrical cavity reinforced by a shell under axisymmetric loads","volume":"2","author":"Zasovenko","year":"2023","journal-title":"New Mater. Technol. Metall. Mech. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1007\/s00707-022-03413-1","article-title":"A general analytical solution of stresses around circular holes in functionally graded plates under various in-plane loading conditions","volume":"234","author":"Khechai","year":"2023","journal-title":"Acta Mech."},{"key":"ref_10","first-page":"1","article-title":"Reduced Models for the Poisson Problem in Perforated Domains","volume":"89","author":"Boulakia","year":"2021","journal-title":"HAL Open Sci. Arch."},{"key":"ref_11","unstructured":"Timoshenko, S.P., and Goodier, J.N. (1970). Theory of Elasticity, McGraw-Hill."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.prostr.2021.10.058","article-title":"The dynamical problem for the infinite elastic layer with a cylindrical cavity","volume":"33","author":"Fesenko","year":"2021","journal-title":"Procedia Struct. Integr."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"116403","DOI":"10.1016\/j.compstruct.2022.116403","article-title":"Minimization of thermal stress in perforated composite plate using metaheuristic algorithms WOA, SCA and GA","volume":"304","author":"Jafari","year":"2023","journal-title":"Compos. Struct."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"104308","DOI":"10.1016\/j.euromechsol.2021.104308","article-title":"Torsion of an elastic half-space with a cylindrical cavity by a punch","volume":"89","author":"Malits","year":"2021","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3449","DOI":"10.1007\/s00419-021-01975-5","article-title":"Scattering of cylindrical inclusions in half space with inhomogeneous shear modulus due to SH wave","volume":"91","author":"Yang","year":"2021","journal-title":"Arch. Appl. Mech."},{"key":"ref_16","unstructured":"Nikolayev, A.G., and Protsenko, V.S. (2011). Obobshchennyy metod Fur\u2019ye v Prostranstvennykh Zadachakh Teorii Uprugosti. Nats. Aerokosm. Universitet im. N.Ye, Zhukovskogo \u201cKHAI\u201d."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1007\/s10958-010-0162-0","article-title":"Apparatus and applications of a generalized Fourier method for transversally isotropic bodies bounded by a plane and a paraboloid of rotation","volume":"171","author":"Nikolaev","year":"2010","journal-title":"J. Math. Sci."},{"key":"ref_18","first-page":"229","article-title":"Some problems for transversaly isotropic space with periodic crack systems","volume":"1","author":"Krainichenko","year":"2024","journal-title":"Probl. Comput. Mech. Strength Struct."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"98","DOI":"10.32620\/reks.2024.2.09","article-title":"The method of determining optimal control of the thermoelastic state of a piece-homogeneous body using a stationary temperature field","volume":"2","author":"Nikolaev","year":"2024","journal-title":"Radioelectron. Comput. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/s10958-016-2974-z","article-title":"Stresses in an infinite circular cylinder with four cylindrical cavities","volume":"217","author":"Nikolaev","year":"2016","journal-title":"J. Math. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1134\/S1995423915020068","article-title":"The first boundary-value problem of the elasticity theory for a cylinder with N cylindrical cavities","volume":"8","author":"Nikolaev","year":"2015","journal-title":"Numer. Anal. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1134\/S0021894416060237","article-title":"Stresses in an elastic cylinder with cylindrical cavities forming a hexagonal structure","volume":"57","author":"Nikolaev","year":"2016","journal-title":"J. Appl. Mech. Tech. Phys."},{"key":"ref_23","first-page":"213","article-title":"Justification of the Generalized Fourier method for the mixed problem of elasticity theory in the half-space with the cylindrical cavity","volume":"2","author":"Protsenko","year":"2016","journal-title":"Visn. ZNU. Phys. Math. Sci."},{"key":"ref_24","first-page":"48","article-title":"Solving a one mixed problem in elasticity theory for half-space with a cy-lindrical cavity by the generalized Fourier method","volume":"2","author":"Ukrayinets","year":"2021","journal-title":"East. Eur. J. Enterp. Technol."},{"key":"ref_25","first-page":"43","article-title":"Investigating a problem from the theory of elasticity for a half-space with cylindrical cavities for which boundary conditions of contact type are assigned","volume":"4","author":"Protsenko","year":"2018","journal-title":"East. Eur. J. Enterp. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1007\/s10778-020-01021-x","article-title":"Stress State of an Elastic Layer with a Cylindrical Cavity on a Rigid Foundation","volume":"56","author":"Miroshnikov","year":"2020","journal-title":"Int. Appl. Mech."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"208","DOI":"10.32347\/2410-2547.2019.103.208-218","article-title":"The study of the first main problem of the theory of elasticity for a layer with a cylindrical cavity","volume":"103","author":"Miroshnikov","year":"2019","journal-title":"Strength Mater. Theory Struct."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Miroshnikov, V., Younis, B., Savin, O., and Sobol, V. (2022). A linear elasticity theory to analyze the stress state of an infinite layer with a cylindrical cavity under periodic load. Computation, 10.","DOI":"10.3390\/computation10090160"},{"key":"ref_29","unstructured":"Pluzhnik, E., and Liubchych, A. (2021). Stress state analysis of a layer with a longitudinal cavity and given non-classical mixed boundary conditions. Science, Theory and Practice, Proceedings of the 29th International Scientific and Practical Conference, Tokyo, Japan, 8\u201311 June 2021, International Science Group."},{"key":"ref_30","unstructured":"Nikichanov, V.V. Determination of the stress state of a layer with a cylindrical cavity under given smooth contact conditions on the layer boundaries and displacements on the cavity surface. Scientific Collection \u201cInterConf\u201d, Proceedings of the 9th International Scientific and Practical Conference \u201cScientific Horizon in the Context of Social Crises\u201d, Tokyo, Japan, 6\u20138 August 2021, Otsuki Press. Available online: https:\/\/ojs.ukrlogos.in.ua\/index.php\/interconf\/issue\/view\/6-8.08.2021\/569."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Miroshnikov, V., Savin, O., Sobol, V., and Nikichanov, V. (2023). Solving the problem of elasticity for a layer with N cylindrical embedded supports. Computation, 11.","DOI":"10.3390\/computation11090172"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Vitaly, M. (2023). Rotation of the layer with the cylindrical pipe around the rigid cylinder. CAMPE 2021: Advances in Mechanical and Power Engineering, Lecture Notes in Mechanical Engineering, Springer.","DOI":"10.1007\/978-3-031-18487-1_32"},{"key":"ref_33","unstructured":"Sverdlov, S. (2025). Determination of the stress-strain state of a bearing connection. Modern Trends in the Development of Economy, Technology and Industry, Collection of Scientific Papers \u201cInternational Scientific Unity\u201d, Proceedings of the 2nd International Scientific and Practical Conference, Toronto, ON, Canada, 12\u201314 February 2025, International Scientific Unity. Available online: https:\/\/isu-conference.com\/en\/archive\/modern-trends-in-the-development-of-economy-technology-and-industry-12-02-25\/."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"44","DOI":"10.15407\/pmach2025.02.044","article-title":"First Main Problem of the Theory of Elasticity for a Layer with Two Thick-Walled Pipes and One Cylindrical Cavity","volume":"28","author":"Denshchykov","year":"2025","journal-title":"J. Mech. Eng. Mash."},{"key":"ref_35","unstructured":"Kosenko, M. (2024, January 5). Solution to an elasticity problem for a layer with cylindrical embedded supports in the form of a cavity and a pipe: Rigid fixation. Proceedings of the Interdisciplinary Scientific and Practical Conference \u201cModern Problems of the Development of the Aerospace Industry of Ukraine: Engineering, Business, Law\u201d, Kharkiv, Ukraine."},{"key":"ref_36","first-page":"8","article-title":"Analysis of the stress state of a reinforced layer with two cylindrical cavities and some contact-type conditions","volume":"19","author":"Ilin","year":"2024","journal-title":"Colloq. J."},{"key":"ref_37","unstructured":"Grebeniuk, I., Kosenko, M., and Ilin, O. (2024, January 17\u201319). Taking into account the loss of adhesion of the reinforcing element with the layer in the calculation of the stress state. Proceedings of the XXXI International Scientific and Practical Conference \u201cScientific Research in the Conditions of Rapid Development of Information Technologies\u201d, Helsinki, Finland. Available online: https:\/\/isu-conference.com\/wp-content\/uploads\/2024\/07\/Scientific_research_in_the_conditions_of_rapid_development_of_information_technologies_July_17_19_2024_Helsinki_Finland.pdf."},{"key":"ref_38","unstructured":"Ilin, O., and Kosenko, M. (2024, January 15\u201317). Solving the problem of elasticity theory for a ball with a circular inclusion and a circular cavity under given boundary conditions of contact type. Proceedings of the XXVIII International Scientific and Practical Conference \u201cDevelopment of Science in the Conditions of Deepening European Integration Processes\u201d, Rome, Italy. Available online: https:\/\/eu-conf.com\/en\/events\/development-of-science-in-the-conditions-of-deepening-european-integration-processes\/."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1016\/j.prostr.2023.12.042","article-title":"FE modelling techniques for the simulation of guided waves inplates with variable thickness","volume":"52","author":"Perfetto","year":"2024","journal-title":"Procedia Struct. Integr."},{"key":"ref_40","unstructured":"Ministry of Regional Development and Construction of Ukraine (2011). DBN V.2.6-98:2009. Konstruktsii Budynkiv i Sporud. Betonni and Zalizobetonni Konstruktsii. State Building Norms V.2.6-98:2009. Structures of Buildings and Erections. Concrete and Reinforced Concrete Structures. Main Provisions, Ukrarkhbudinform."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/11\/270\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,17]],"date-time":"2025-11-17T15:58:51Z","timestamp":1763395131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/11\/270"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,17]]},"references-count":40,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["computation13110270"],"URL":"https:\/\/doi.org\/10.3390\/computation13110270","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,17]]}}}