{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:57:07Z","timestamp":1760057827917,"version":"build-2065373602"},"reference-count":52,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T00:00:00Z","timestamp":1740960000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Fapesp (Sao Paulo State Research Foundation), Brazil","award":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"],"award-info":[{"award-number":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"]}]},{"name":"CNPq (Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico), Brazil","award":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"],"award-info":[{"award-number":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"]}]},{"DOI":"10.13039\/501100001665","name":"ANR (Agence Nationale Recherche) in France","doi-asserted-by":"publisher","award":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"],"award-info":[{"award-number":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004795","name":"IUF (Institut Universitaire France)","doi-asserted-by":"publisher","award":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"],"award-info":[{"award-number":["2023\/16272-1","2022\/15644-0","2020\/13362-1","313046\/2021-2","ANR-20-CE46-0012-01","1479"]}],"id":[{"id":"10.13039\/501100004795","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Kinematically exact rod models were a major breakthrough to evaluate complex frame structures undergoing large displacements and the associated buckling modes. However, they are limited to the analysis of global effects, since the underlying kinematical assumptions typically take into account only cross-sectional rigid-body motion and ocasionally torsional warping. For thin-walled members, local effects can be notably important in the overall behavior of the rod. In the present work, high-fidelity simulations using elastic 3D-solid finite elements are employed to provide input data to train a Deep Neural Newtork-(DNN) to act as a surrogate model of the rod\u2019s constitutive equation. It is capable of indirectly representing local effects such as web\/flange bending and buckling at a stress-resultant level, yet using only usual rod degrees of freedom as inputs, given that it is trained to predict the internal energy as a function of generalized rod strains. A series of theoretical constraints for the surrogate model is elaborated, and a practical case is studied, from data generation to the DNN training. The outcome is a successfully trained model for a particular choice of cross-section and elastic material, that is ready to be employed in a full rod\/frame simulation.<\/jats:p>","DOI":"10.3390\/computation13030063","type":"journal-article","created":{"date-parts":[[2025,3,3]],"date-time":"2025-03-03T09:04:49Z","timestamp":1740992689000},"page":"63","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A DNN-Based Surrogate Constitutive Equation for Geometrically Exact Thin-Walled Rod Members"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9543-1406","authenticated-orcid":false,"given":"Marcos Pires","family":"Kassab","sequence":"first","affiliation":[{"name":"Department of Structural and Geotechnical Engineering, Polytechnic School, University of S\u00e3o Paulo, P.O. Box 61548, S\u00e3o Paulo 05424-970, SP, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6770-9634","authenticated-orcid":false,"given":"Eduardo de Morais Barreto","family":"Campello","sequence":"additional","affiliation":[{"name":"Department of Structural and Geotechnical Engineering, Polytechnic School, University of S\u00e3o Paulo, P.O. Box 61548, S\u00e3o Paulo 05424-970, SP, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6502-0198","authenticated-orcid":false,"given":"Adnan","family":"Ibrahimbegovic","sequence":"additional","affiliation":[{"name":"Laboratoire Roberval, Centre de Recherche Royallieu, University of Technology Compi\u00e8gne\u2014Alliance Sorbonne University, Rue de Docteur Schweitzer, Hauts-de-France, 60200 Compi\u00e8gne, France"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/BF01602645","article-title":"On one-dimensional finite-strain beam theory: The plane problem","volume":"23","author":"Reissner","year":"1972","journal-title":"Z. Angew. Math. Phys. Zamp"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/0045-7825(82)90069-X","article-title":"An excursion into large rotations","volume":"32","author":"Argyris","year":"1982","journal-title":"Comput. 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