{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:06:12Z","timestamp":1760148372895,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,4,24]],"date-time":"2023-04-24T00:00:00Z","timestamp":1682294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT, Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia, through IDMEC under LAETA","award":["UIDB\/50022\/2020"],"award-info":[{"award-number":["UIDB\/50022\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>Depending on the load level, structures can experience a material nonlinearity known as elastoplasticity, which has an important role in the behaviour of structures. In order to avoid the elastoplastic behaviour, it is necessary to find the optimal thickness distribution, which corresponds to the minimum mass that provides an elastic behaviour for a certain load level. The elastoplasticity analysis of functionally graded axisymmetric shells under axisymmetric mechanical loading, and the subsequent optimization, was performed by using a simple conical frustum finite element model with two nodal circles; three degrees of freedom per node, which was based on Kirchhoff\u2019s theory allowing for shear deformation; and using a reduced numerical integration procedure that is essential for its success when applied to thin shells. The formulation accounts for the calculation of the displacements and through-thickness stress distribution, including the effective stress. In this work, the thickness was the design variable in the optimization procedure and the mass was the objective function that needed to be minimized subject to a constraint imposed on the effective stress. The optimization solutions were obtained by using a feasible arc interior point gradient-based algorithm. Some illustrative examples were performed, and the corresponding results are presented and discussed.<\/jats:p>","DOI":"10.3390\/app13095325","type":"journal-article","created":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T02:06:35Z","timestamp":1682388395000},"page":"5325","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Optimization on Elastoplasticity of Functionally Graded Shells of Revolution under Axisymmetric Loading"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2529-583X","authenticated-orcid":false,"given":"Jos\u00e9","family":"Sim\u00f5es Moita","sequence":"first","affiliation":[{"name":"IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"given":"Aur\u00e9lio Lima","family":"Ara\u00fajo","sequence":"additional","affiliation":[{"name":"IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5159-5361","authenticated-orcid":false,"given":"Victor","family":"Franco Correia","sequence":"additional","affiliation":[{"name":"Escola Superior N\u00e1utica Infante D. Henrique, Av. Eng. Bonneville Franco, 2770-058 Pa\u00e7o de Arcos, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4645-138X","authenticated-orcid":false,"given":"Crist\u00f3v\u00e3o Mota","family":"Soares","sequence":"additional","affiliation":[{"name":"IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal"}]},{"given":"Jos\u00e9","family":"Herskovits","sequence":"additional","affiliation":[{"name":"COPPE\/UFRJ, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-909, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/S0997-7538(99)80011-4","article-title":"Axisymmetric bending of functionally graded circular and annular plates","volume":"18","author":"Reddy","year":"1999","journal-title":"Eur. J. Mech. A\/Solids"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.ijsolstr.2007.07.023","article-title":"Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk","volume":"45","author":"Li","year":"2008","journal-title":"Int. J. 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